Constructing units in commutative group rings.

Klaus Hoechsmann at Vancouver. *

0. Introduction. For any finite abelian group A, let U(A) denote the multiplicative group of the integral group ring ZZA. In his 1965 paper [1], Bass showed how to construct generators for a free abelian subgroup of finite index in U(A). More recently, another such construction was described in [7], with the advantage of producing subgroups of much lower index in U(A), but the disadvantage of being constrained to p-groups A of odd order. The aim of this paper is to extend the second procedure to arbitrary A, and to clarify its relation to the first one.

The Bass units are obtained by a kind of surrogate “division” within ZZA of (x − 1)^{m} by (y − 1)^{m}, where x ∈ A and y ∈ A each generate the same cyclic group C ⊆ A, and m > 1 is an appropriate integer. The procedure of [7] depends on first constructing a suitable integral polynomial w(X), and then using the set of units {w(z) | z ∈ A} to generate a subgroup of U(A). For odd p-groups A, this sometimes yields “all” units (modulo the trivial units ±A), and quite often “almost all” (especially for regular p). By contrast, the finite index of the Bass construction tends to be huge. For instance, if |A| = 67, this index is greater than 10^{56}, while the other method still gives all units.

In the present paper, we start with a cyclic group C. Putting G = Aut (C), and fixing a generator x of C, we construct a canonical G-map w : Δ^{2}(G) ⟶ U(C), where Δ^{2}(G) is the square of the kernel Δ(G) of the obvious ring surjection (coefficient sum): ZZG ⟶ ZZ. In fact, if H denotes G modulo its subgroup of order 2, our map induces an isomorphism

w : Δ^{2}(H) ⟶^{∼} W (C) ⊂ U(C),

whose image W (C) is our basic building block. The group B_{m}(C) generated by the Bass units attached to C happens to be the w-image of the submodule mΔ(H) ⊆ Δ^{2}(H). The unit w(x) which plays the lead role in [7] is simply the image of a generator for the principal ideal Δ^{2}(H) in the p-group case. As C ranges over all cyclic subgroups of a given A, the product of the W (C) — which is direct — forms a free subgroup of finite index in U(A).

The basic isomorphism Δ^{2}(H) ≃ W (C) can be described by explicit formulas, making it a routine matter to exhibit a basis of each W (C). Does this method ever yield all units if n = |A| is not a prime power? The answer seems to be affirmative for n < 74. The shortfall at n = 2 × 37 is not directly related to the irregularity of 37: similar ones occur at n = 5 × 17, or (to stay with n = 2p) at n = 2 × 197, for instance. Work is in progress on the relatively easy cases of n = 2p^{ν} and n = 3p^{ν}, yet even there it is difficult to see consistent patterns. The evidence to date is too patchy for a verdict, but at this modest level of sophistication there seems to be nothing better, so far, for obtaining subgroups of reasonably small index.

The techniques used in this paper are elementary. Apart from easy abelian character theory, they involve the use of the standard exponential type homomorphism from the additive group of Δ(A) to A — which gets into the act for both A = C and A =Aut (C) — as well as the ring ZZ[x]/(1+x+⋯+x^{n−1}) of “generic” n-th roots of 1. The main work consists in modifying a few well-known ingredients to make them fit together. The heaviest pieces of back-ground equipment are the cyclic induction theorem of Bass-Milnor and the cylotomic independence theorem of Bass. Both of these can ultimately be linked to results of Dirichlet: the theorem on units and the non-vanishing of L(1, χ).

* Work supported, in part, by a grant from NSERC of Canada.

The first two sections accomplish the assembling of the tools and their deployment in the construction. Section 3 makes the connection with the general but small unit groups of [1], on the one hand, and the large but specialized ones of [7], on the other. Section 4 summarizes what is known for p-groups, and extends most of it to the case p = 2, at the same time providing some new and shorter proofs. The last two sections go off in almost opposite directions: Section 5 looks back at the main construction and ties up some loose ends, while Section 6 begins the exploration of units for abelian groups of twice odd order.

Thanks are due to Antonio Giambruno and the Algebra Group at the University of Palermo for their hospitality, their helpful comments, and their patience in listening to a first muddled version of this paper; and to Jürgen Ritter of Augsburg University for showing the way to the examples given at the end of it.

1. Preliminaries. In this section, we recall four well-known results, reformulating them so as to fit into our scheme. We sketch their proofs as much as seems necessary for the narrative. At the same time we introduce some notation.

With A as in the introduction, consider the ring involution ⋆ : ZZA ⟶ ZZA, which is induced by mapping each element of A to its inverse (i.e. z^{⋆} = z^{−1} for z ∈ A). A unit u ∈ U(A) will be called symmetric if u^{⋆} = u, anti-symmetric if u^{⋆} = u^{−1}. The subgroups of symmetric and anti-symmetric units will be labelled by U^{+}(A) and U^{−}(A), respectively.

Clearly U^{−}(A) is just the torsion subgroup of U(A), because in each (complex) Wedderburn component of ZZA the elements of finite multiplicative order are characterized by the condition |w|^{2} = w\overline{w} = 1. This is the key to the following theorem of G. Higman [5].

Lemma 1.1: The torsion subgroup of U(A) is U^{−}(A) = ±A. Proof: ±A ⊆ U(A)_{tor} = U^{−}(A), by the preceding remark. On the other hand, it is easy to show that U^{−}(A) = ker [v ↦ vv^{⋆}] ⊆ ±A. Indeed, if v = ∑_{z∈A} a_{z}z ∈ ZA, the coefficient of identity in vv^{⋆} is ∑_{z} a_{z}^{2} ≠ 1, unless v = ±z.

Next, we want a functorial decomposition of U(A) into U(A)_{tor} and a free ZZ-module. Getting a split into U^{−}(A) and U^{+}(A) would be ideal — and that is precisely what happens if A has odd order. In general, however, the complementary subgroup has to be defined differently.

Let Δ(A) be the kernel of the obvious ring homomorphism ZZA ⟶ ZZ (“augmentation”), and define the subgroups U_{i}(A) ⊂ U(A) by U_{i}(A) = U(A) ∩ (1 + Δ^{i}(A)), for i = 1, 2. Since U(A) = ±U_{1}(A), we may as well restrict our attention to U_{1}(A). The latter turns out to be a direct product of the finite group A by the torsion-free group U_{2}(A). This is the gist of Lemma 2.6 of Cliff-Sehgal-Weiss [3], whose proof we now review, because we shall later need the explicit component maps of this splitting.

We start with the standard exponential type homomorphism

e_{A} : ∑_{z∈A} c_{z}z ⟼ ∏_{z∈A} z^{c_{z}}  (1)

from the additive group of ZZA to the multiplicative group A ⊂ U_{1}(A). When there is no risk of confusion, we shall leave off the subscript, and label this map simply e. It satisfies the important congruence e(δ) ≡ 1 + δ mod Δ^{2}(A), which is valid for all δ ∈ Δ(A), and is easily derived by iterating the elementary identity (yz − 1) − (y − 1) − (z − 1) = (y − 1)(z − 1). In particular, e(δ) = 1 ⟺ δ ∈ Δ^{2}(A), and e induces an isomorphism Δ(A)/Δ^{2}(A)⟶^{∼}A. But it does more.

Lemma 1.2: (i) On U_{1}(A) the map e_{A} is multiplicative and yields a split exact sequence

1 ⟶ U_{2}(A) ⟶ U_{1}(A)⟶^{e}A ⟶ 1.  (2)
(ii) Moreover U_{2}(A) ⊂ U^{+}(A).

Proof: e will map (1 + δ_{1})(1 + δ_{2}) = (1 + δ_{1} + δ_{2} + δ_{1}δ_{2}) into e(δ_{1} + δ_{2}) = e(δ_{1})e(δ_{2}). This establishes multiplicativity. Exactness and splitting are obvious, since ker(e) = U_{2}(A) by the remark preceding the lemma, and e(z) = z for z ∈ A by definition. This proves (i).

For (ii) note that u ∈ U(A) ⟹ v = u/u^{⋆} ∈ U_{1}^{−}(A) = A by Lemma 1.1. If u ∈ U_{2}(A), this means u = u^{⋆}, because A ∩ U_{2}(A) = {1} by part (i).

Remarks:

1. For most purposes it suffices to remember the resulting direct decompositions

U_{1}(A) = A × U_{2}(A)  and  U_{1}^{+}(A) = A_{2} × U_{2}(A),  (3)

where A_{2} ⊆ A is the subgroup generated by elements of order 2. For groups A of odd order this shows that U_{2}(A) = U_{1}^{+}(A).

2. It is sometimes useful to consider the group U_{11}(A) (subscript to be read as “one-one”), consisting of those units which lie in the kernel of all characters ι : A ⟶{±1}, i.e. characters of A/A^{2}. Since such a ι maps every (z_{1}−1)(z_{2}−1) into 0 or −4, it can never take the value −1 on 1+Δ^{2}(A), and hence trivializes U_{2}(A). In view of (ii) above, this yields the inclusion U_{2}(A) ⊆ U_{11}^{+}(A) which, by (3), is proper if and only if A_{2} ∩U_{11}(A) is non-trivial. Since A ∩ U_{11}(A) = A^{2}, this amounts to saying that U_{2}(A) = U_{11}^{+}(A) ⟺ A_{2} ∩ A^{2} = {1}, i.e. iff A has no elements of order 4.

Our third lemma deals with a cyclic group C of order n > 2, its automorphism group G = Aut (C) = (ZZ/nZZ)^{×}, and the maximal order M_{C} within the rational group algebra QC. In the Wedderburn decomposition of the latter, M_{C} can be identified with ⨁_{d} ZZ[ζ_{d}], where d denotes divisors of n, and ζ_{d} a primitive d-th root of 1. Fixing a generator x of C, we obtain an obvious map

u : Δ(G) ⟶ U_{1}(M_{C}) by δ ⟼ [ψ(x) − 1]^{δ},  (4)

with ψ : C ⟶ C^{×} running over the non-trivial characters of C. The following result is due to W. Franz [4], Section 9.

Lemma 1.3: Let δ = ∑_{σ} a_{σ}σ ∈ Δ(G) be such that a_{σ} = a_{⋆σ} for all σ ∈ G. Then

u(δ) = 1 implies δ = 0. Proof: For any system {b_{τ} | τ ∈ G} of complex numbers, and any character χ : G ⟶ C^{×}, a routine computation shows

∑_{σ} a_{σ}χ(σ) · ∑_{τ} b_{τ} χ(τ^{−1}) = ∑_{ρ} χ(ρ^{−1}) ∑_{σ} a_{σ}b_{ρσ}  (5)

with ρ, σ, τ ranging over G. Put b_{σ} = log |ζ^{σ} − 1|, where ζ ≠ 1 is an n-th root of 1 yet to be chosen. Then u(δ) = 1 implies ∏_{σ}(ζ^{ρσ} − 1)^{a_{σ}} = (ψ(x) − 1)^{δ} = 1, where ψ(x) = ζ^{ρ}. Hence the right hand side of (5) is zero for any choice of ζ.

If δ ≠ 0, however, there must be a χ ≠ 1 such that ∑_{σ} a_{σ}χ(σ) ≠ 0 and χ(⋆) = 1. But by choosing ζ so as to be primitive with respect to the conductor f_{χ}, we also get ∑_{τ} χ(τ^{−1}) log |ζ^{τ} −1| ≠ 0. Indeed this expression equals −f_{χ}L(1, χ)/τ(χ), involving L-functions and Gauss sums in the usual notation of analytic number theory (cf. Washington’s book [8], Theorem 4.9, p. 37).

While Franz’s Lemma injects Δ_{⋆}(G) = {δ ∈ Δ(G) | ⋆δ = δ} into U_{1}(M_{C}) via u, Bass’s Independence Theorem (cf. [1], Theorem 3, p. 401) simultaneously involves all the analogous maps Δ_{⋆}(G_{K}) ⟶ U_{1}(M_{K}) ⊆ U_{1}(M_{C}), where K ⊆ C runs over all non-trivial subgroups: they jointly define an injection of the direct sum ⨁_{K} Δ_{⋆}(G_{K}) into U_{1}(M_{C}). But we need not stop with cyclic groups.

Lemma 1.4: For any finite abelian group A, there is an injection of the direct sum

⨁_{K⊆A} Δ_{⋆}(G_{K}) ⟶ U_{1}(M_{A}),  (6)

where K ⊆ A runs over all non-trivial cyclic subgroups, and the components

Δ_{⋆}(G_{K}) ⟶ U_{1}(M_{K}) ⊆ U_{1}(M_{A}) are defined as in (4). Proof: For every K of order > 2, the rank of Δ_{⋆}(G_{K}) is \frac{1}{2}|G_{K}| − 1, which equals the free rank of the units in the ring of |K|-th roots of 1. Since there is a one-to-one correspondence between cyclic subgroups and factor groups of A, the left hand term of (6) has the same rank as the right hand term taken modulo torsion, by Dirichlet’s Unit Theorem. Hence the lemma is equivalent to saying that the map (6) has an image of finite index.

By a theorem of Bass and Milnor (cf. [2], Ch. XI, Th. 7.1.c), the product of the inclusions U_{1}(C) ↪ U_{1}(A), as C ⊆ A runs over all cyclic subgroups, does have finite index. Hence so does the corresponding product of inclusions U_{1}(M_{C}) ↪ U_{1}(M_{A}). Combining this with Bass’s Independence Theorem (the case A = C of the present lemma) yields the stated result.

The reader wishing to look up details in [1] should be warned that Bass does not nail down his version of u to U_{1}(M_{C}) but defines it via (ζ − 1)^{δ} on a free-floating non-trivial n-th root of 1. This makes for some awkwardness when it comes to the inclusion K ↪ C. On the other hand, it does not require fixing a generator in every cyclic subgroup, as does the presentation given here.

2. The Construction. To construct a standard subgroup of finite (and potentially computable) index in U_{1}(A), we begin by taking A = C to be cyclic of oder n. Fixing a generator x of C, we have ZZC = ZZ[x] with x^{n} − 1 = 0. Let ξ be the image of x in the quotient ring ZZ[ξ] = ZZ[x]/(1 + x + ⋯ + x^{n−1}), the “n-th generic cyclotomic ring”.

It is important to realize that ξ is not simply a primitive n-th root of 1 (unless n is prime), but a kind of generic one, capable of specializing to a d-th root for any 1 ≠ d | n. In fact, Q[ξ] is the direct sum of Q[ζ_{d}], for all d ≠ 1 and with ζ_{d} denoting a primitive d-th root; i.e. Q[ξ] is essentially QC without the component corresponding to the trivial character. Unlike QC or Q[ζ_{n}], it does not contain its d-th counterparts, for d | n (although, like QC, it always maps onto them).

Note that ZZC is a fiber product of ZZ[ξ] by ZZ as shown by the diagram

ZZC ⟶ ZZ
↓  ↓
ZZ[ξ] ⟶^{ε} ZZ/nZZ

(7)

where ε means working modulo (ξ − 1), i.e. taking the coefficient sum modulo n. Indeed, given a ∈ ZZ and ∑_{i} a_{i}ξ^{i} ∈ ZZ[ξ] such that ∑_{i} a_{i} = a + kn, we construct the unique ancestral element ∑_{i} a_{i}ξ^{i} − k(1 + x + ⋯ + x^{n−1}) ∈ ZZC.

Whereas (x − 1) is a zero divisor in QC, its image (ξ − 1) is a unit in Q[ξ]. Hence it makes perfect sense to write (ξ − 1)^{σ−1} for any σ ∈ G = Aut (C), and more generally (ξ − 1)^{δ} for any δ ∈ Δ(G). Since

(ξ − 1)^{σ−1} = \frac{ξ^{c} − 1}{ξ − 1} = 1 + ξ + ⋯ + ξ^{c−1},  (8)

where c > 0 is prime to n and σ : x ↦ x^{c}, these elements actually lie in ZZ[ξ]. Moreover they are units there, because (ξ − 1)^{1−σ} = (ξ^{σ} − 1)^{τ−1} with τ = σ^{−1}. Hence we obtain a natural G-homomorphism

u : Δ(G) ⟶ U(ξ) by u : δ ⟼ (ξ − 1)^{δ},  (9)

where U(ξ) denotes the unit group of ZZ[ξ]. The obvious injection U(ξ) ↪ U_{1}(M_{C}) reconciles this definition with the one given in (4).

Remark: We note in passing that (ξ − 1)^{α} with α ∈ ZZG is a unit only if α ∈ Δ(G), except in certain trivial cases all with n | 12. Indeed, α = a + δ with a ∈ ZZ, and if one can map (ξ − 1) homomorphically to the non-unit (ζ_{q} − 1), where q > 4 is a prime power dividing n, it follows that a = 0 and α = δ.

Lemma 2.1: The G-map u yields a morphism of short exact sequences

\begin{aligned}1 & ⟶ & Δ^{2}(G) & ⟶ & Δ(G) & ⟶^{e} & G & ⟶ & 1 \\ & & ↓ & & u ↓ & & ↓ \\ 1 & ⟶ & U_{1}(C) & ⟶ & U(ξ) & ⟶^{ε} & U(n) & ⟶ & 1, \end{aligned}  (10)

where U(n) = (ZZ/nZZ)^{×} = G, and the third vertical arrow represents identity.

Proof: By the remarks preceding Lemma 1.2, Δ^{2}(G) equals the kernel of the surjection e_{G} : Δ(G) ⟶ G. By the pull-back property of (7), an element of U_{1}(C) is uniquely determined by the pair 1 ∈ ZZ and v = ∑_{i} a_{i}ξ^{i} ∈ ZZ[ξ] such that ∑_{i} a_{i} ≡ 1 mod n, i.e. v ∈ ker(ε). So much for exactness.

The commutativity of the right hand square in (10) can be checked on a typical generator δ = (σ − 1) ∈ Δ(G): if x^{σ} = x^{c} with c > 0, we have u_{δ}(ξ) = 1 + ξ + ⋯ + ξ^{c−1}; this goes to c ∈ (ZZ/nZZ)^{×} under ε, which is the same as e(σ − 1) = σ ∈ G — QED.

The image of the restricted map u : Δ^{2}(G) ⟶ U_{1}(C) is the basic ingredient for constructing units in U_{1}(C). However, to avoid torsion, we shall still have to subject it to the canonical projection e′ : U_{1}(C) ⟶ U_{2}(C) corresponding to the split U_{1}(C) = C × U_{2}(C) given by Lemma 1.2.

Theorem 2.2: The composite map

Δ^{2}(G)⟶^{u}U_{1}(C)⟶^{e′}U_{2}(C)  (11)
induces an injection w : Δ^{2}(H) ⟶ U_{2}(C), where H = G/ < ⋆ >.

Proof: To prove that e′ ∘ u factors over the canonical map h : Δ^{2}(G) ⟶ Δ^{2}(H) induced by taking G modulo < ⋆ >, we must show that u maps ker(h) = (⋆ − 1)Δ(G) into ker(e′) = C. This follows from the trivial calculation

(ξ − 1)^{⋆−1} = \frac{ξ^{−1} − 1}{ξ − 1} = −ξ^{−1},  (12)

by which u takes (⋆ − 1) into a torsion element. Therefore u : (⋆ − 1)Δ(G) ⟶ U_{1}(C)_{tor}.

Injectivity of w follows from Lemma 1.3, which grants this property to the restriction of u to the H-module Δ_{⋆}(G) = (⋆ + 1)Δ(G) ⊂ Δ(G), consisting of all δ ∈ Δ(G) such that δ = ⋆δ. The canonical homomorphism h clearly bijects Δ_{⋆}(G) onto 2Δ(H) ⊂ Δ(H).

If an element α of Δ^{2}(H) lies in the kernel of w, then so does 2α, which equals h(δ) for suitable δ ∈ Δ_{⋆}(G). Since w ∘ h = e′ ∘ u, it follows that e′(u_{δ}) = 1 and therefore u_{δ} ∈ C. This implies nδ = 0 because u is injective on Δ_{⋆}(G), whence δ = 0 and finally 2α = h(δ) = 0. So α = 0, proving that w is injective, as claimed.

The following proposition gives explicit formulas for u and w in terms of the sums s_{i}(x) = 1+x+⋯+x^{i−1} defined for any i > 0. Without risk of confusion, we use the same notation for elements of G and their counterparts in H.

Proposition 2.3: Let α = (σ − 1)(τ − 1) be a typical generator of Δ^{2}(G) with σ : x ↦ x^{c}, and b, k be positive integers with bc = 1 + kn. Then

u_{α}(x) = s_{b}(x^{σ})s_{c}(x^{τ}) − ks_{n}(x)  and  w_{α}(x) = x^{−(τ−1)(c−1)/2}u_{α}(x).  (13)

Proof: As given above, u_{α}(x) is easily seen to satify u_{α}(1) = 1 and u_{α}(ξ) = (ξ − 1)^{α}, since α = (σ^{−1} − 1)σ + (σ − 1)τ. To project this into the torsion-free component U_{2}(C), we must divide it by e_{C}(u_{α}(x)). Applying (1) to s_{c}(x^{τ})s_{b}(x^{σ}) = ∑_{i,j} x^{jτ+iσ} we get ∏_{i,j} x^{jτ+iσ}, with 0 ≤ i, j < b, c, which yields x raised to the power τbc(c − 1)/2 + σcb(b − 1)/2. This amounts to τ(c − 1)/2 + σ(b − 1)/2, because x^{(c−1)/2} is always well-defined, and therefore bc = 1 + kn may be dropped from the exponent. Finally, taking into account the term −ks_{n}(x), a straight-forward computation yields the desired result.

Remark: All this could be greatly simplified in case |C| is odd. Then x^{1/2} makes sense, and we could replace u by the map v : Δ(G) ⟶ U^{+}(ξ) which takes δ ∈ Δ(G) directly into the ⋆-symmetric unit (ξ^{1/2} − ξ^{−1/2})^{δ}. As in Lemma 2.1, we would get a morphism of short exact sequences

\begin{aligned}1 & ⟶ & Δ^{2}(G) & ⟶ & Δ(G) & ⟶^{e} & G & ⟶ & 1 \\ & & w ↓ & & v ↓ & & ↓ \\ 1 & ⟶ & U_{1}^{+}(C) & ⟶ & U^{+}(ξ) & ⟶^{ε} & U(n) & ⟶ & 1, \end{aligned}  (14)

by the same reasoning, because ε ∘ v = ε ∘ u. For every δ ∈ Δ(G), we would have v_{δ}(ξ) = ξ^{−δ/2}u_{δ}(ξ) by definition, and w would be obtainable from v by simple restriction to Δ^{2}(G). There would be no need to mention the projection e′ : U_{1}(C) ⟶ U_{2}(C), which would be identity on U_{1}^{+}(C) anyway. For α = (σ − 1)(τ − 1) ∈ Δ^{2}(G), and b, c, k as in Proposition 2.3, we would have the simple formula

w_{α}(x) = v_{b}(x^{σ})v_{c}(x^{τ}) − ks_{n}(x)  with  v_{c}(x) = x^{\frac{1−c}{2}} s_{c}(x).  (15)

This fails for even n because (c − 1)/2 is known only modulo n/2 if c is given modulo n. Note, however, that v can always be defined as a mere ZZ-map by simply picking one (of at most two) interpretations of x^{(1−c)/2} for every basis element (σ − 1) ∈ Δ(G). Restricted to 2Δ(G), this would always yield the same G-map v : 2Δ(G) ⟶ U^{+}(ξ).

Definition 2.4: The image W (C) := w(Δ^{2}(H)) will be be referred to as the group of well-formed units belonging to C. Unlike w itself, W (C) is independent of the choice of generator x for C. For any finite abelian A, let Y (A) denote the product ∏_{C} W (C) ⊆ U_{2}(A), with C ⊆ A ranging over all cyclic subgroups (of order > 2); it will be called the group of constructible units of A.

Just as Theorem 2.2 was an adaptation of Lemma 1.3 (Franz), so the next theorem will be a recasting of Lemma 1.4 (Bass-Milnor).

Theorem 2.5: For any finite abelian group A, the product

Y (A) = ∏_{C⊆A} W (C)  (16)

is direct and has finite index in U_{2}(A).

Any homomorphism A ⟶ A′ of abelian groups maps Y (A) into Y (A′) by surjecting each W (C) onto W (C′), where C′ denotes the image of C.

Proof: (13) shows that w_{α}(x)^{2} = w_{α}(x)w_{α}(x^{−1}) = u_{α}(x)u_{α}(x^{−1}) = u_{α+⋆α}(x). Therefore the square of any constructible unit y = ∏_{C} w_{C} is in the in the image of the injection (6) shown in Lemma 1.4, and y = 1 implies w_{C} = 1 for all C. The finiteness of the index follows by the same rank count as was used in the proof of Lemma 1.4.

For the second statement, let x′ ∈ C′ be the image of the generator x of C. Every automorphism σ : x ↦ x^{c} of C can also be seen as an automorphism σ′ : x′ ↦ (x′)^{c} of C′, and in that sense every α ∈ Δ^{2}(H) produces an α′ ∈ Δ^{2}(H′). Since the correspondence α ↦ α′ is surjective, and since every w_{α}(x) maps to w_{α′}(x′), the surjectivity W (C) ⟶ W (C′) follows.

3. Some Comparisons. The aim of this section is twofold: to relate the groups W (C) to the kind of unit group actually constructed by Bass in [1], and to show that they agree with the procedure described in [7] for p-groups. Our notation will be as follows: A is finite abelian, C ⊆ A a maximal cyclic subgroup, G = Aut (C), and H = G/ < ∗ >.

For our first topic, we stay in the cyclic context with C =< x >. If m is any multiple of the group order |G|, we obviously have U(ξ)^{m} ⊆ ker(ε) = U_{1}(C), and hence u mapping mΔ(G) onto a subgroup B_{m}(C) ⊂ U_{1}(C). This group of “Bass units of level m” has the special generators b_{m}(x, σ) = u(σ − 1)^{m} (corresponding to 1 ≠ σ ∈ G), each of which is characterized as being the unique element of U_{1}(C) satisfying the equation

(x^{σ} − 1)^{m} = (x − 1)^{m}b_{m}(x, σ).  (17)

Obviously b_{ml}(x, σ) = [b_{m}(x, σ)]^{l} for any integer l > 0. To facilitate reference to [1], we should point out that Bass works with a fixed m and uses the symbol [e(x^{σ})/e(x)] instead of b_{m}(x, σ).

Remembering that m is bound to be even, we can also consider the group B_{m}^{+}(C) of “symmetrized Bass units of level m”, which is generated by the units

b_{m}^{+}(x, σ) = x^{(1−σ)m/2}b_{m}(x, σ).  (18)

Characterized by the equations [x^{−m/2}(x − 1)^{m}]^{σ} = [x^{−m/2}(x − 1)^{m}]b_{m}^{+}(x, σ) analogous to (17), these units clearly satisfy b_{m}^{+}(x^{−1}, σ) = b_{m}^{+}(x, σ), so that B_{m}^{+}(C) ⊆ U_{1}^{+}(C). Using the map v : 2Δ(G) ⟶ U^{+}(ξ) mentioned in the remark preceding Definition 2.4, we can alternatively write B_{m}^{+}(C) = v(mΔ(G)).

In [1], the level m is taken to be a multiple of both |G| and n = |C|. Since (c, n) = 1 implies x^{(1−c)n/2} = 1 irrespective of the parity of n, we always have b_{m}^{+}(x, σ) = b_{m}(x, σ) in this case, and the distinction between B_{m}(C) and B_{m}^{+}(C) disappears. This simplifies the discussion at the price of exponentially decreasing the size of the groups so obtained. We shall here take a more conservative approach and use B_{m}^{+}(C) throughout, but keep in mind that B_{m}^{+}(C) = B_{m}(C) for values of m as envisaged in [1].

Proposition 3.1: If m is a multiple of (2, n)|G|, then B_{m}^{+}(C) = w(mΔ(H)).

In particular,

B_{m}^{+}(C) ⊂ W (C)  and  [W (C) : B_{m}^{+}(C)] = \frac{m^{|H|−1}}{|H|} > 1.  (19)

Proof: We first prove that B_{m}^{+}(C) ⊆ U_{2}(C). Since U_{1}^{+}(C) = C_{2} × U_{2}(C), there is nothing more to show if n = |C| is odd. In the even case, m is a multiple of 2|G|, and B_{2|G|}^{+}(C) = [B_{|G|}^{+}(C)]^{2} ⊆ U_{2}(C) as well.

Since mΔ(G) ⊆ Δ^{2}(G), we now conclude that B_{m}^{+}(C) = v(mΔ(G)) is contained in v(Δ^{2}(G)) ∩ U_{2}(C) = u(Δ^{2}(G)) ∩ U_{2}(C) ⊆ W (C). In fact, for any δ ∈ Δ(G) the units u_{2δ}(x) and v_{2δ}(x) coincide modulo torsion, hence must be equal if they are in U_{2}(A). It follows that the bijection Δ^{2}(H)⟶^{w}W (C) maps the subgroup mΔ(H) isomorphically onto B_{m}^{+}(C). Therefore the index [W (C) : B_{m}^{+}(C)] can be computed by comparing [Δ(H) : Δ^{2}(H)] with [Δ(H) : mΔ(H)], whence the result.

Remark: Of course, Y (A) contains the direct product X_{f}(A) of Bass groups B_{f(K)}^{+}(K), where K ⊆ A ranges over cyclic subgroups and f(K) is an appropriate value for m — say (2, |K|) times the order of Aut (K). It is easy to see that X_{f}(A) will not be functorial in A unless the level m is allowed to vary. However, there is no way of choosing f so as to make X_{f}(A) preserve surjections, as Y (A) does.

Before narrowing our focus all the way down to p-groups A, let us see what ensues when the group H happens to be cyclic. This is clearly so for |C| = p^{ν} (for p = 2 the generator being x ↦ x^{3}), but also for |C| = 2p^{ν} and in certain cases for |C| = 3p^{ν}. One of the shortcomings of [7] is its failure to recognize the role of Δ^{2}(H). The following proposition hinges on the fact that the ideals Δ(H) and Δ^{2}(H) are principal if H is cyclic. It generalizes the first part of Theorem 3.5 in [7] and, together with the agreement between Eq.(15) above and formula (‡) on p.326 (loc. cit.), shows that the notion of “constructible unit” is the same in both papers.

Proposition 3.2: Suppose that H is cyclic, and let α be a generator of the ideal Δ(H).

Then Y (A) is generated over ZZ by the H-set {w_{α}(z) | z ∈ A}, which can be reduced

to ZZ-basis of Y (A) by omitting one element from each of its H-orbits. Proof: If σ ∈ H is a generator, multiplication by (σ − 1) induces isomorphisms

ZZH/Σ(H)⟶^{∼}Δ(H)  and  Δ(H)⟶^{∼}Δ^{2}(H),  (20)

where Σ(H) denotes the ideal generated by the sum over H. Mapping 1 to w_{α}(x), we therefore have an H-isomorphism of ZZH/Σ(H) with W (C) by Theorem 2.2. Hence the product over the H-orbit {w_{α}(z) | < z >= C} is trivial, and a basis of W (C) is obtained from this orbit by throwing out any one of its elements.

By the second part of Theorem 2.5, the corresponding element w_{α}(z) can similarly serve to generate W (K), where K = < z > for any z ∈ A. Since every z ∈ A occurs exactly once as a generator of one of the cyclic subgroups K ⊆ A, we thus obtain each of the components of the direct product Y (A) = ∏_{K} W (K).

4. Prime Powers. We shall now see how some of the other results of [7] fit into the general scheme described above, and how they extend to 2-groups. As in the earlier paper, we first deal with the problem of lifting cyclotomic units.

A special feature of p-groups (this time an exclusive one) is the faithfulness with which well-formed units are portrayed in the highest Wedderburn component. The reason is that, for cyclotomic units,a primitive p^{ν}-th root of 1 behaves like the “generic” ξ introduced in Section 2 above.

Lemma 4.1: Let |C| = n > 2 be a prime power, and ζ denote a primitive n-th root of unity. Consider the maps φ : ξ ↦ ζ and ψ : x ↦ ζ of U(ξ) and U(C) into U(ζ), respectively. Then

(a) φ yields an injection of the group < ξ >[1 − ξ]^{Δ(G)} into U(ζ) ;

(b) for δ ∈ Δ(G), we have (1 − ζ)^{δ} ∈ ψ(U_{1}(C)) ⟺ δ ∈ Δ^{2}(G).

Proof: For each μ = 1, …, ν let ψ_{μ} : C ⟶ C^{×} be the p^{ν−μ}-th power of ψ, and F_{μ} be the ψ_{μ}-image of QC. It is well known and elementary that ψ_{μ}(x) and 1 − ψ_{μ}(x) are the F_{ν}/F_{μ}-norms of ζ and 1 − ζ, respectively (note: we have taken ψ_{ν}(x) = ζ). Hence all Wedderburn components of any element of the form ξ^{a}[1 − ξ]^{δ} are determined by its ψ-image ζ^{a}[1 − ζ]^{δ}. So much for (a).

For (b), imagine a (1 − ζ)^{δ} ∈ ψ(U_{1}(C)). By (a), it has in U(ξ) exactly one pre-image, namely (1 − ξ)^{δ}, which must therefore be liftable to an element of U_{1}(C). This does not mean that we know δ itself, but it does mean that ε((1 − ξ)^{δ}) = 1 — cf. diagram (10) in Lemma 2.1. Hence it follows that e_{G}(δ) = 1, and δ ∈ Δ^{2}(G), as was to be shown.

Notation: For any n > 2 (not just prime powers), and any n-th root of unity ζ ≠ 1 (not just primitive ones), we put

U^{⊕}(ζ) = < ζ >[1 − ζ]^{Δ(G)} ∩ I.R^{×}.  (21)

These are the only elements of ZZ[ζ] which have a hope of turning up as Wedderburn components of constructible units. If A is abelian with A^{n} = {1}, we write U^{⊕}(A) for the group of all units in U(A) which are mapped into the appropriate U^{⊕}(ζ) by every character (ring surjection) ZZA ⟶ ZZ[ζ]. Such units are called “circular”. Note that all constructible units are automatically circular, i.e. Y (A) ⊆ U^{⊕}(A).

At first sight, circular units may look too special to be interesting. However, their cyclotomic images are so well-explored as to make them a natural object of study. For small values of n, they include all of U_{1}^{+}(A), and, in general, their index in that group can be expressed in terms of class-numbers of real cyclotomic fields.

Returning to the p-group context of Lemma 4.1, we now describe the ψ-image of W (C), which turns out to encompass all “liftable” elements of U^{⊕}(ζ). The following Proposition 4.2 covers essentially the same ground as Theorems 1.3 and 2.3 of [7], but has a much simpler proof. It does omit the p-adic part of the first of these theorems, but in exchange it includes the case p = 2. Unfortunately the latter burdens its statement with an annoying reference to the almost always trivial subgroup C_{2} ⊂ C of order (n, 2).

Proposition 4.2: If n = |C| is a prime power and ζ a primitive n-th root of 1, the map ψ : x ↦ ζ induces a bijection

C_{2} · W (C) ⟶^{∼} ψ(U_{1}(C)) ∩ U^{⊕}(ζ).  (22)

Proof: Only the surjectivity might need some comment. Consider a t ∈ U_{1}(C) such that ψ(t) ∈ U^{⊕}(ζ). Then, by the proof of Lemma 4.1, all Wedderburn components of t are real, and hence t ∈ U_{1}^{+}(C). By Lemmas 2.1 and 4.1 (b), it is also in C · u(Δ^{2}(G)). This means that t ∈ U_{1}^{+}(C) ∩ u(Δ^{2}(G)) = C_{2} · W (C).

According to the preceding proposition, W (C) is as large as it can be: it contains all elements of U_{1}^{+}(C) whose ψ-images are cyclotomic units (up to sign, if p = 2). At the end of this section, we quote a theorem of [7] (there numbered 3.5), which claims a similar kind of maximality for Y (A) in case A is a p-group for odd regular p. But first we need an important intermediate result.

Proposition 4.3: If A is a finite abelian p-group, then so is U_{1}^{⊕}(A)/Y (A).

Proof: Let c(A) denote the index of A_{2} · Y (A) in U_{1}^{⊕}(A). One of the achievements of [7] is showing that c(A) < ∞ without recourse to the Bass Independence Theorem, i.e., without analytic input other than what is required for Dirichlet’s Unit Theorem. The inductive mechanism which accomplishes this (first for A = C, as usual) also serves to establish the other claims made about c(A), and is contained in the proof of Lemma 2.5 (loc. cit.). A careful reading of the latter reveals that it actually amounts to deriving a recursion equation of the form

c(C) = c(C^{p}) · i(C^{p}),  (23)

which, however, remains implicit in [7]. For any non-trivial subgroup K ⊆ C, the integer i(K) is defined as follows. Let π : ZZK ⟶ ZZK^{p} be the map induced by taking p-th powers in K, and remember that t^{π} and t^{p} are congruent modulo p for any t in the group ring. Hence t ↦ t^{π−p} = t^{π}/t^{p} produces a homomorphism from U(K) into the kernel of the reduction map U(K) ⟶ UIF_{p}K. The factor i(K) in (23) turns out to be the index of this homomorphism restricted to the units of interest here, i.e.

U_{1}^{⊕}(K) ⟶^{π−p} ker [U_{1}^{⊕}(K) ⟶ UIF_{p}K].  (24)

With this definition of i(K), the right hand side of (23) is just the index of the counterclockwise composite in diagram (9) of [7]. To see that the clockwise index of that diagram equals c(C), one needs two identifications. The first of these uses the direct product decomposition U_{1}^{⊕}(C) = C_{2}W (C) × (ker ψ) and a modification of the obvious splitting C_{2}Y (C) = C_{2}W (C) × ∏_{i>0} W (C^{p^{i}}) to represent U_{1}^{⊕}(C)/C_{2}Y (C) as a factor group of ker ψ. The second one is the isomorphism between ker ψ and the kernel shown in (24), which is brought about by the representation of ZZC as a fiber product of ZZ[ζ] by ZZC^{p} over IF_{p}C^{p}.

By means of (23), a simultaneous induction on c(K), i(K), and the index of U_{1}^{⊕}(K)⟶^{π}U_{1}^{⊕}(K^{p}) then shows these three integers to be p-powers, cf. Lemma 2.6 (loc. cit.). The extension to non-cyclic A is based on a non-trivial but general fact about Bass-Milnor indices, which applies equally to any prime p (cf. [7], Section 3).

Proposition 4.4: For a finite p-group A of odd order, Y (A) ≠ U_{1}^{⊕}(A) if and only if

p is irregular and |A| > p. Proof: (Sketch) It is clear from the proof of the previous result, that everything hinges on the triviality of i(K) for cyclic subgroups K ⊆ A. The surjectivity of (24) for odd regular p (as well as its failure for irregular p) may be viewed as generalizing Kummer’s lemma about p-th roots of cyclotomic units. The regularity of p is used to link U_{1}^{+}(K) to a group of units (⋆-symmetric with augmentation and G-norm = 1) in the p-adic group-ring ZZ_{p}K, where the study of (π − p) is relatively easy.

Remarks:

1. If p = 2, this requires modification owing to the fact that A_{2} ⊂ U_{1}^{⊕}(A). For instance, if A = C has order 8, then Y (C) = W (C) ≃ Δ^{2}(H) is infinite cyclic, while C_{2}W (C) is not. In fact, taking b = c = 3 and σ = τ in our Proposition 2.3, we get the generator w_{α}(x) = (2x^{4} − 1) + (x + x^{−1}) − (x^{3} + x^{−3}) for W (C). Now x^{4}w_{α}(x) cannot be in W (C), although it is certainly in U_{1}^{⊕}(C) and shows no superficial symptoms of non-constructibility until one evaluates e_{C}.

The proof that c(A) = 1 proceeds exactly as outlined above, until it comes to the surjectivity of (24). Multiple changes are necessary in the fairly long chain of p-adic arguments which enter at this stage, because U_{2}(A) is no longer U_{1}^{+}(A), and division by 2 is no longer allowed. A more detailed account would drastically shift the emphasis of this paper, and will be omitted.

2. If p satisfies Vandiver’s Conjecture (as all p < 125000 are known to do), the index [U_{1}(A) : U_{1}^{⊕}(A)] is prime to p. Therefore U_{1}^{⊕}(A)/Y (A) is the Sylow p-subgroup (i.e. the essential part) of U_{1}(A)/Y (A). By Proposition 2.1 and the remark following it, this would be false if Y (A) were replaced by a product of Bass groups such as the X_{f}(A) described there. So would be the analogs of statements to the effect that U_{1}^{⊕}(A) = Y (A) whenever p > 2 is regular, that U_{1}^{⊕}(C) = W (C) for |C| = p, or that U_{1}^{+}(C) = W (C) for p ≤ 67.

With p as above, the main result of [6] equates c(A) to the order of a certain canonical group of ideal classes in ZZA, thereby giving it some intrinsic status despite its ad hoc appearance.

5. A Variant. With α = (σ − 1)(τ − 1), the explicit formulas (13) can be seen as defining 2-cocycles ũ(σ, τ) = u_{α}(x) and w̃(σ, τ) = w_{α}(x) with values in U_{1}(C) and U_{2}(C), respectively. The first of these clearly corresponds to the extension U(ξ) of U_{1}(C) by G. In this section we look at the extension of U_{2}(C) by H given by the second. At the same time, we examine an analogue of the fundamental diagram (10) based directly on the group H.

We begin with a partial generalization of Higman’s Theorem (Lemma 1.1 above), partial because it is restricted to U^{+}(C) for a cyclic goup C = < x > of order n, generalization because it deals with torsion modulo the ideal Σ(C) ⊂ ZZC generated by (1 + x + ⋯ + x^{n−1}).

Lemma 5.1: Let t ∈ ZZC be such that t = t^{⋆}.

Then t^{2} ≡ 1 mod Σ(C) implies t ≡ ±z mod Σ(C) with z ∈ C_{2}.

Proof: If t = ∑_{i} a_{i}x^{i}, the hypotheses say that a_{i} = a_{n−i} for i = 1, …, [n/2] and that

( ∑_{i=0}^{n−1} a_{i}x^{i})^{2} = ∑_{i=0}^{n−1} c_{i}x^{i}  with  c_{k} = c_{0} − 1  (25)

for k = 1, …, n − 1. Since we are working mod Σ(C), and since the sign of t does not matter, we may moreover assume that a_{0} = 0 and a_{r} > 0, where a_{r} marks the first non-zero coefficient, that is, a_{0} = ⋯ = a_{r−1} = 0 but a_{r} ≠ 0. We then claim:

a_{j+kr} = \{ \begin{matrix}0 , & for 0 < j < r; \\ 1 , & for j = r; \end{matrix}  (26)

as long as j + kr < n. To prove this claim, we define an n-tuple {b_{i}} by

b_{i} = \{ \begin{matrix}0 , & for 0 < i < r; \\ a_{i−r} , & for r < i < n; \end{matrix}  (27)

but b_{r} = a_{n−r} and b_{0} = 0. (Except for that last switch, these are the coefficients of tx^{r}.) The usual convolution formula for t · t^{⋆} now brings the coefficients c_{r} and c_{0} into the act:

∑_{i=0}^{n−1} a_{i}b_{i} = a_{r}^{2} + ∑_{i=r}^{n−1} a_{i}a_{i−r} = a_{r}^{2} + c_{r} = a_{r}^{2} − 1 + c_{0}.  (28)

Since c_{0} = ∑_{i} a_{i}^{2}, and {b_{i}} is just a reshuffling of {a_{i}}, the Cauchy-Schwarz Lemma says a_{r}^{2} − 1 + c_{0} ≤ c_{0} and allows equality only if b_{i} = a_{i} for all i (remember: b_{r} = a_{r}). We conclude that a_{r} = 1 and a_{i} = a_{i−r} for r < i < n. Rewriting this as a_{j+r} = a_{j} for 0 < j < n − r, and noting that r ≤ n − r (because t^{⋆} = t), we obtain the formula (26) by iteration, and the claim is proved.

It is easy to see that r divides n. Indeed, suppose we had n = s + qr with 0 < s < r. Since n ≥ 2r and hence q ≥ 2, we would have n − r = s + (q − 1) r fitting the pattern j + kr < n of (26) with 0 < j < r, and get the impossible result a_{n−r} = 0.

Two possibilties now emerge: either r > 1 or r = 1. In the first case, (26) guarantees that adjacent coefficients of t are never both non-trivial. Hence c_{1} = ∑_{i>0} a_{i}a_{i−1} = 0, whence c_{0} = 1, and all but one a_{i} are zero, so that r = n/2 and t = x^{r}. In the second case, 1 = a_{1} = ⋯ = a_{n−1} and t = −1+(1+x+⋯+x^{n−1})

— QED.

In terms of the n-th generic cyclotomic ring ZZ[ξ], Lemma 5.1 can be summarized by the formula

U^{+}(ξ)_{tor} = ±< ξ >_{2},  (29)

or by saying that the quotient group V^{+}(ξ) = U^{+}(ξ)/ ± < ξ >_{2} is torsion free. Taking a cue from diagram (14) we now define a map v̇ : Δ(H) ⟶ V^{+}(ξ) which will yield w by restriction to Δ^{2}(H). Unlike u, this map is injective.

Proposition 5.2: Let δ = (σ − 1) be a typical generator of Δ(G) with σ : x ↦ x^{c}.

Then δ ↦ v_{c}(ξ) = ξ^{(1−c)/2}s_{c}(ξ) defines an H-injection v̇ : Δ(H) ⟶ V^{+}(ξ) of ZZ-free

ZZH-modules, which induces a morphism of exact sequences

\begin{aligned}1 & ⟶ & Δ^{2}(H) & ⟶ & Δ(H) & ⟶^{e} & H & ⟶ & 1 \\ & & w ↓ & & v̇ ↓ & & ↓ \\ 1 & ⟶ & U_{2}(C) & ⟶ & V^{+}(ξ) & ⟶^{ε} & V (n) & ⟶ & 1, \end{aligned}  (30)

where V (n) = U(n)/{±1}. Proof: Applying ⋆ to s_{c}(ξ) = (ξ^{c} − 1)/(ξ − 1) produces ξ^{1−c}s_{c}(ξ). Therefore v_{c}(ξ) ∈ U^{+}(ξ). Moreover, its image in V^{+}(ξ) depends only on the automorphism σ : x ↦ x^{c}. In fact, if b = c + kn, the quotient v_{c}(ξ)/v_{b}(ξ) is the 2-torsion element ξ^{kn/2}. Therefore the correspondence (σ − 1) ↦ v_{c}(ξ) actually defines a map Δ(G)⟶^{v̇}V^{+}(ξ). Since v_{n−1}(ξ) = −ξ^{−n/2}, this map is trivial on (⋆ − 1), and hence is definable on Δ(H).

To obtain (30), we turn back to (14) but regard it as a diagram of ZZ-maps (which is all the central v can be for even n). Working modulo 2-torsion turns this into (30), because U_{1}^{+}(C) = C_{2} × U_{2}(C) and ε(−1) = −1, and because the ad hoc ZZ-map v changes into the well-defined v̇. The injectivity of v̇ follows from that of w, because Δ^{2}(H) has finite index in the free ZZ-module Δ(H), and U_{2}(C) injects into V^{+}(ξ). However, it can also be deduced more directly by comparing v̇ with the injection u : Δ_{⋆}(G) ↪ U^{+}(ξ).

6. Double Odd Order. As a first foray into groups of even order, consider A^{♯} = A × B with |B| = 2 and |A| odd. Let B = < y >, and let φ : y ↦ 1 and ψ : y ↦ −1 denote the characters of B. Then we have the following fiber product of rings:

ZZA^{♯} ⟶^{ψ} ZZA
φ ↓  ↓
ZZA ⟶ IF_{2}A.

(31)

Indeed, the general element ∑_{z}(a_{z} + a′_{z}y)z of ZZA^{♯} (with z running over A) is mapped to ∑_{z}(a_{z} ± a′_{z})z by φ and ψ, respectively. Given a pair of components ∑_{z} b_{z}z and ∑_{z} c_{z}z such that b_{z} ≡ c_{z} (mod 2), we obviously get unique integers a_{z} and a′_{z} such that a_{z} + a′_{z} = b_{z} and a_{z} − a′_{z} = c_{z}, for all z ∈ A.

We may therefore think of U(A^{♯}) as the subgroup U(A) ×_{2} U(A) ⊂ U(A) × U(A) consisting of all pairs (u, v) which agree modulo 2. The natural subgroup U(A) ⊂ U(A^{♯}), from the inclusion A ↪ A^{♯}, shows up as the diagonal in this representation. Writing (u, v) as the product of (u, u) and (1, v/u), we see that U(A^{♯}) is actually the direct product U(A) × ker_{2} U(A), where ker_{2} X denotes ker [X ⟶ UIF_{2}A], for any subgroup X ⊆ U(A). The component projections of this split are φ and γ : t ↦ ψ(t)/φ(t), respectively.

Of course, we are not as interested in all of U(A^{♯}) as in the subgroup U_{2}(A^{♯}) = U_{11}^{+}(A^{♯}) — cf. Remark 2 following Lemma 1.2 — which is just the fiber product U_{1}^{+}(A) ×_{2} U_{1}^{+}(A). Restricting the preceding description to this subgroup, we again get a direct product

U_{11}^{+}(A^{♯}) ≃ U_{1}^{+}(A) × ker_{2} U_{1}^{+}(A),  (32)

with the component maps φ and γ, respectively. This relation remains in force if U^{+} is replaced by U^{⊕} throughout.

Remark: It follows that U_{11}^{+}(A^{♯}) = U_{11}^{⊕}(A^{♯}) ⟺ U_{1}^{+}(A) = U_{1}^{⊕}(A).

Our main task is to identify the constructible units in terms of this decomposition. It will turn out that Y (A^{♯}), too, is a direct product with the same projections φ and γ.

Proposition 6.1: The pair of maps φ, γ yields an isomorphism

Y (A^{♯}) ≃ Y (A) × Y (A)^{τ−2},  (33)

where τ ∈ Aut (A) is given by τ : z ↦ z^{2}.

The proof hinges on the following lemma.

Lemma 6.2: Let K ⊆ A be cyclic, and put K^{♯} = K × B. Then γ yields a surjection

W (K^{♯}) ⟶ W (K)^{τ−2}.

Proof: It suffices to show that γ(w) = φ(w)^{τ−2}, or equivalently ψ(w)φ(w) = φ(w)^{τ}, for any w ∈ W (K^{♯}). This means that w(−ζ)w(ζ) = w(ζ^{2}) for all |K|-th roots ζ of 1. Now, if ξ is a generic 2|K|-th root of unity, we know that w(ξ) = ξ^{a}(ξ − 1)^{α} with suitable a ∈ ZZ and α ∈ Δ^{2}(G), where G = Aut (K^{♯}). Let us compute w(−ζ)w(ζ) one factor at a time.

Since w(−1) = 1 implies that a ∈ 2ZZ, we get (−ζ)^{a}ζ^{a} = ζ^{2a} alright. A typical factor of (ξ − 1)^{α} looks like (ξ − 1)^{σ−1} with σ : ξ ↦ ξ^{c}, where c is prime to 2|K| and hence odd. The lemma thus follows from the simple identity

((−ζ)^{c} − 1)(ζ^{c} − 1) = −(ζ^{2c} − 1).  (34)

Proof of 6.1: To establish Eq.(33), we remember that Y (A^{♯}) = ∏_{K} (W (K) · W (K^{♯})) as K runs over all cyclic subgroups of A, and show that

W (K) · W (K^{♯}) ⟶^{∼} W (K) × W (K)^{τ−2},  (35)

with component projections φ and γ as before. We need only worry about surjectivity. Given v ∈ W (K), let w ∈ W (K^{♯}) be such that φ(w) = v. Then (v · 1) ∈ W (K) · W (K^{♯}) is mapped to (v, 1) by the pair φ, γ. According to the lemma, we have γ(w) = v^{τ−2}. Hence (v^{−1} · w) ∈ W (K) · W (K^{♯}) is mapped to (1, v^{τ−2}) by the pair φ, γ, and we are done.

Taken together, the equations (32) and (33) immediately yield an isomorphism of finite groups:

\frac{U_{11}^{⊕}(A^{♯})}{Y (A^{♯})} ≃ \frac{U_{1}^{⊕}(A)}{Y (A)} × \frac{ker_{2} U_{1}^{⊕}(A)}{Y (A)^{τ−2}}.  (36)

This is the formula we have been aiming for. Again the circles around the subsript +-signs are optional. We are mainly interested in the following special case. Corollary 6.3: U_{1}^{⊕}(A) = Y (A) ⟹ U_{11}^{⊕}(A^{♯})/Y (A^{♯}) ≃ [ ker_{2} Y (A)]/ Y (A)^{τ−2}.

From here on in, we specialize much further by taking A = C to be of prime order p > 3. In that case, we know that U_{1}^{⊕}(C) = Y (C) = W (C), which is a monogenic H-module isomorphic to Δ^{2}(H) ≃ Δ(H) ≃ ZZH/Σ(H) by Eq.(20). Since τΣ(H) = Σ(H), we have (2 − τ)Σ(H) = Σ(H) and therefore

W (C)/W (C)^{τ−2} ≃ Δ^{2}(H)/(2 − τ)Δ^{2}(H) ≃ ZZH/(2 − τ)ZZH.  (37)

The order of this is det(2 − M_{τ}), where M_{τ} is the h × h-matrix representing multiplication by τ in ZZH, and of course h = (p − 1)/2. If τ has order f | h in H, this matrix is a direct sum of g = h/f identical f × f submatrices corresponding to the actions of τ on the cosets of H modulo < τ >. This gives the first of the following two equalities:

[W (C) : W (C)^{τ−2}] = (2^{f} − 1)^{g} = [U_{1}^{+}IF_{2}C : 1].  (38)

To see the second equality, remember that IF_{2}C is semisimple, and that τ is the Frobenius automorphism on the constituent fields; then apply a little Galois lore. By semi-simplicity, we have

IF_{2}C ≃ IF_{2} ⊕ IF_{2}[ζ_{p}]  and  U_{1}^{+}IF_{2}C ≃ UIF_{2}[θ_{p}],  (39)

where θ_{p} = ζ_{p} + ζ_{p}^{−1}, and IF_{2}[θ_{p}] is the reduction modulo 2 of the real cyclotomic ring ZZ[θ_{p}]. The Galois group of the latter is H, and the decomposition group of 2 therein is < τ >. Hence the ideal generated by 2 splits into g prime ideals of inertial degree f (the order of the Frobenius automorphism), and accordingly IF_{2}[θ_{p}] is the direct sum of exactly g Galois fields of 2^{f} elements. This proves (38).

Using the abbreviation im_{2}X = im [X ⟶ UIF_{2}A] for subgroups X ⊆ U(A), we now divide (38) by the obvious relation [W (C) : ker_{2} W (C)] = [im_{2}W (C) : 1] to obtain

[U_{11}^{⊕}(C^{♯}) : Y (C^{♯})] = [U_{1}^{+}IF_{2}C : im_{2}W (C)],  (40)

by Corollary 6.3, provided that |C| = p > 3 as stipulated. In keeping with the notation of Section 4, this index will be labelled c(C^{♯}).

Examples: For simplicity, we consider only those p for which 2 generates IF_{p}^{×}. Then IF_{2}[θ_{p}] is a field, and U_{1}^{+}IF_{2}C ≃ IF_{2}[θ_{p}]^{×} is cyclic of order 2^{h} − 1. Since the ideal Δ^{2}(H) is generated by α = (τ − 1)^{2}, the H-module W (C) is generated by w_{α}(x), and the subgroup im_{2}W (C) ⊆ IF_{2}(θ_{p})^{×} is generated by w_{α}(ζ_{p}) = (ζ_{p}^{1/2} − ζ_{p}^{−1/2})^{α} = θ_{p}^{τ^{−1}α} = θ_{p}^{τ^{−1}}, because (τ − 1) fixes IF_{2}(θ_{p})^{×}. Therefore c(C^{♯}) = [IF_{2}(θ_{p})^{×} : < θ_{p} >] in these cases.

1. If 3 < p ≤ 541 with 2 generating IF_{p}^{×}, this index is 1, and hence U_{11}^{⊕}(C^{♯}) = Y (C^{♯}), except when

p = 37, 101, 197, 269, 349, 541.

For 3 < p ≤ 67 and p ≠ 37, we actually get all of U_{11}^{+}(C^{♯}) constructible, because then U_{1}^{⊕}(C) = U_{1}^{+}(C), since the class number h_{p}^{+} is 1 — cf.[8], p.352. It should be kept in mind that Y (C^{♯}) = W (C) × W (C^{♯}), which is isomorphic to Δ^{2}(H)×Δ^{2}(H), has a very simple and explicit description in terms of generators and relations.

2. Since the prime factorization of 2^{h} − 1 is not entirely trivial, neither is the computation of the exact order (mod 2) of θ_{p} or the exact value of c(C^{♯}). For the first five of the exceptional primes listed above, the smallest (and probably only) prime divisor of c(C^{♯}) was found to be 3; for the last one, p = 541, it is 7. Thus the arithmetical nature of c(A^{♯}) appears more erratic than that of c(A) for p-groups A, which is always a p-power.

References.

[1] H. Bass, The Dirichlet unit theorem, induced characters, and Whitehead groups of finite groups, Topology 4 (1966), 391 – 410.

[2] H. Bass, Algebraic K-Theory, Benjamin, New York (1968).

[3] G. H. Cliff, S. K. Sehgal, A. R. Weiss, Units of integral group rings of metabelian groups, J. of Alg. 73 (1981), 167 – 185.

[4] W. Franz, Über die Torsion einer Überdeckung, J. reine u. angew. Math. 173 (1935), 245 – 254.

[5] G. Higman, The units of group rings, Proc. London Math. Soc. (2) 46 (1940), 231 – 248.

[6] K. Hoechsmann, Local units and circular index in abelian p-group rings, J. Pure and Appl. Alg. (to appear).

[7] K. Hoechsmann, J. Ritter, Constructible units in abelian p-group rings, J. Pure and Appl. Alg. 68 (1990) 325 – 339.

[8] L. Washington, Introduction to cyclotomic fields, Springer, New York (1982)

Department of Mathematics, University of British Columbia, Vancouver, V6T 1Y4, Canada.