Alternating Sums in Integral Group Rings.

Klaus Hoechsmann, Department of Mathematics, University of B.C., Vancouver, Canada, V6T 1Y4. and S. K. Sehgal, Department of Mathematics, University of Alberta, Edmonton, Canada, T6G 2G1.

Abstract: In this note, we exhibit generators up to finite index of UZZA, the unit group of the integral group ring ZZA of a finite abelian p-group A, for infinitely many (in fact, over half of all) primes p. These generators have the simple form 1 − z + z^{2} − ⋯ + z^{c−1}, for z ∈ A and c odd, relatively prime to the order of z. They obviously lie in the group U_{1}ZZA of units with augmentation one. If −2 is a primitive root modulo p, these generators work not only for p-groups A but also for extensions of the latter by elementary abelian 3-groups.

1. Preparations. Let C =< x > be cyclic of order n. For any odd integer positive c, consider the element

g_{c}(x) = ∑_{i=0}^{c−1}(−x)^{i} ∈ ZZC  (1)

of the group ring. The oddness of c makes the augmentation g_{c}(1) equal to 1; the corresponding expressions for even values of c would have zero augmentation, and therefore are uninteresting for the study of units.

We wish to investigate the usefulness of these g_{c}(x) for producing units of ZZC. Our first observation is that g_{c}(x) cannot be a unit if n is even, since in that case x may be mapped to −1, and g_{c}(−1) = c > 1. Therefore we may assume that n is odd. In particular, τ : x ↦ x^{2} defines an automorphism of C.

Next we observe that it is necessary for c to be relatively prime to n. Otherwise we can find an n-th root ζ ≠ 1 of unity such that ζ^{c} = 1. Setting x = ζ in the obvious identity

(1 + x)g_{c}(x) = 1 + x^{c},  (2)

we then get g_{c}(ζ) = 2(1 + ζ)^{−1}, a non-unit in ZZ[ζ]. The fact that (1 + ζ)^{−1} ∈ ZZ[ζ] is most easily see by taking the τ-norm of both sides of the identity (1 + ζ)(1 − ζ) = (1 − ζ^{2}), which gives N_{τ} (1 + ζ) = 1. Let us summarize.

Lemma 1:

g_{c}(x) is a unit in ZZC if and only if n is odd and (c, 2n) = 1.

The obvious fact that g_{c}(x) = g_{c+2n}(x), allows restricting the indices to 1 ≤ c < 2n, with (c, 2n) = 1 — which represent the elements of the group G = Aut (ZZ/2nZZ) = Aut (C). Hence we may use the notation g_{c}(x) = g_{σ}(x) with σ ∈ G. Recalling the elementary identity

f_{a}(X)f_{b}(X^{a}) = f_{ab}(X),  (3)

valid in the polynomial ring ZZ[X] for the partial sums f_{a}(X) = 1 + X + ⋯ + X^{a−1} and any positive integers a, b, and noting that g_{c}(x) = f_{c}(−x), we get the important cocycle relation

g_{σ}(x)g_{ρ}(x^{σ}) = g_{σρ}(x).  (4)

Since g_{1}(x) = 1, this gives a formula for g_{σ}(x)^{−1}. Moreover it shows that the group of units generated by the set {g_{σ}(x) | σ ∈ G} is a G-module depending only on C.

For the sequel, it is important to relate this group A(C) to a potentially larger group W(C) constructed as follows. Let ZZ[ξ] denote the polynomial ring ZZ[X] modulo the ideal generated by f_{n}(X), and note that f_{c}(ξ) depends on c only modulo n, thus permitting the notation f_{σ}(ξ) = f_{c}(ξ) for σ ∈ G with σ : ξ ↦ ξ^{c}. Again the cocycle relation (3) shows that the set {f_{σ}(ξ) | σ ∈ G} generates a subgroup W(ξ) of the unit group UZZ[ξ], and that W(ξ) = W(ξ^{ρ}) for any ρ ∈ G. Now, UZZ[ξ] contains U_{1}ZZC as the kernel of the natural “augmentation” map aug_{n} : UZZ[ξ] ⟶ U(ZZ/nZZ) = G, because the obvious diagram

ZZC ⟶ ZZ[ξ]
↓  ↓
ZZ ⟶ ZZ/nZZ

(5)

of ring homomorphisms is a fibre-product. We put W(C) = W(ξ)∩U_{1}ZZC. In other words, W(C) is the kernel of aug_{n} on W(ξ).

The relation between A(C) and W(C) is most easily stated in terms of the homomorphism

u_{ξ} : ΔZZG ⟶ UZZ[ξ]  given by  u_{ξ}(σ − 1) = f_{σ}(ξ),  (6)

which is a G-map because f_{σ}(ξ) can be written as (ξ −1)^{σ−1} in Q[ξ]. The cocycle relation (3) ensures that the image of any G-submodule of ΔZZG remains unchanged when ξ is replaced by any G-conjugate ξ^{ρ}.

Lemma 2:

We have W(C) = u_{ξ}(Δ^{2}ZZG) and A(C) = u_{ξ}((τ − 1)ΔZZG). Proof: Since aug_{n} : W(ξ) ⟶(ZZ/nZZ)^{×} is surjective, the index [W(ξ) : W(C)] equals the order of G. But so does [ΔZZG : Δ^{2}ZZG]. Hence the first assertion of the lemma means that u_{ξ}(Δ^{2}ZZG) is contained in U_{1}ZZC, in other words, that aug_{n}(f_{σ}(ξ)^{ρ−1}) = 1 for any ρ ∈ G — which is obvious.

The second assertion then follows from the equation

g_{σ}(ξ) = (1 + ξ)^{σ−1} = f_{σ}(ξ)^{τ−1},  (7)

which results from interchanging exponents in f_{2}(ξ)^{σ−1} = (ξ − 1)^{(τ−1)(σ−1)}. By (7), the generating set {g_{σ}(x) | 1 ≠ σ ∈ G} of A(C) is the u_{ξ}-image of the generating set {(τ − 1)(σ − 1) | 1 ≠ σ ∈ G} of (τ − 1)ΔZZG.

2. Finite Index. In this section we shall characterize the finite abelian groups A, with A^{n} = 1, for which the set {g_{c}(z) | z ∈ A, (c, 2n) = 1} generates a subgroup of finite index in UZZA. We start by considering a cyclic group C of order n, as in Section 1, and the unit groups A(C) and W(C) associated with it. Let ⋆ ∈ G be the automorphism given by x ↦ x^{−1}, and T be the group generated by τ in H = G/ < ⋆ >. Furthermore, let i_{n} denote the index [H : T ].

Proposition 1:

The corank of A(C) in W(C) is i_{n} − 1. If i_{n} = 1, then A(C) ≡ W(C) and has

the basis {g_{c}(x) | 1 < c < n, (c, 2n) = 1}, modulo torsion.

Proof: Since (ξ − 1)^{⋆−1} = −ξ^{−1} is of finite order, u_{ξ} maps (⋆ − 1)Δ(G) into the torsion part of U_{1}ZZC, and hence induces an H-homomorphism

w_{x} : Δ^{2}(H) ⟶ U̇_{1}ZZC,  (8)

where U̇ means U modulo torsion. This new map is injective by an old lemma of W. Franz [3] — cf. also [4], Lemma 1.3 and Theorem 2.2. Since the corank in question may be computed in U̇_{1}ZZC, it equals that of the inclusion (τ − 1)Δ(H) ↪ Δ^{2}(H), because of Lemma 2 above.

Since |T |(τ−1) ∈ Δ^{2}(T) ⊆ (τ−1)Δ(H), the latter group is of finite index in (τ−1)ZZH. But this is the kernel of ZZH ⟶ ZZ[H/T ]. Hence (τ − 1)Δ(H) has corank [H : T ] in ZZH, and one less in Δ(H) or Δ^{2}(H). If it is zero, i.e. T = H, then obviously (τ − 1)Δ(H) = Δ^{2}(H), and A(C) = W(C).

Clearly {(τ − 1)(σ − 1) | 1 ≠ σ ∈ T } is a basis for Δ^{2}(T). If T = H, its image under w_{x} is the set indicated in the proposition, because the conditions 1 < c < n and (c, 2n) = 1 select exactly one element from every pair σ, ⋆σ.

Remark: Instead of fixing the generator x of C and varying the subscript c in g_{c}(x), we can obtain a basis for A(C) modulo torsion by proceeding in the opposite fashion: fix an odd number b such that ρ : x ↦ x^{b} generates H, and consider the set of units {g_{b}(x^{c}) | 1 < c < n, (c, 2n) = 1}, provided again that T = H. Indeed, letting σ in (4) take on the values ρ^{k}, for k = 1, …, |H| − 1, we have g_{ρ}(x^{σ}) = g_{ρ^{k+1}}(x)/g_{ρ^{k}}(x). This set generates A(C) since g_{1}(x) = 1.

Lemma 3:

If A is a finite abelian group, the product Y(A) = ∏_{C} W(C), where C ⊆ A runs over all cyclic subgroups, is of finite index in U_{1}ZZA.

Proof: With “dot” again meaning “modulo torsion”, the product over the maps (8) yields an injection

⨁_{C⊆A} Δ^{2}(H_{C}) ⟶ U̇_{1}ZZA  (9)

of finite index. This follows almost immediately from the cyclotomic independence theorem of Bass [1] and the cyclic induction theorem of Bass-Milnor (ibid.) — cf. also [4], Lemma 1.4 and Theorem 2.5. Consequently Ẏ(A) has finite index in U̇_{1}ZZA. Since the torsion subgroup of U_{1}ZZA is finite (namely, A), so does Y(A) in U_{1}ZZA.

We remark in passing that, unlike its image in U̇_{1}ZZA, the product Y(A) = ∏_{C} W(C) is neither direct nor are its constituents W(C) torsion-free.

Theorem:

Let A be a finite abelian group of odd period n. The product Y_{alt}(A) = ∏_{C} A(C), where C ⊆ A ranges over all cyclic subgroups, is of finite index in U_{1}ZZA if and only if i_{n} = 1, in which case Y_{alt}(A) ≡ Y(A) modulo torsion.

Proof: Modulo torsion, Y(A) is the direct product of the groups W(C), by the injectivity of (9). Proposition 1 now shows that Y_{alt}(A) has corank ∑_{C}(i_{C} −1) in Y(A). Since |C| = n for some C, the condition i_{n} = 1 is necessary for Y_{alt}(A) to be of finite index. Since m | n clearly implies i_{m} | i_{n}, it is also sufficient.

Corollary:

If i_{n} = 1, let b be an odd number such that x ↦ x^{b} generates H. Then the set {g_{b}(z) | z ∈ A} generates the group Y(A) modulo torsion.

Proof: Obviously, z ↦ z^{b} generates H_{K} for every cyclic subgroup K =< z > of A. If we let z range just over the generators K, the resulting units g_{b}(z) are more than enough to generate W(K) modulo torsion, by the remark following Proposition 1.

3. Arithmetical Conditions. For odd n, the condition i_{n} = 1 can occur only when n is either a prime power or of the form n = 3p^{k}, where p is a prime > 3. In all other cases, the group H_{n} = G_{n}/ < ⋆ > is manifestly non-cyclic. To discuss the two admissible cases, we put q = p^{k} and n = 3q, with the understanding that (q, 6) = 1 in the second instance. Then G_{q} = (ZZ/qZZ)^{×} is cyclic, but G_{n} ≃ G_{3} × G_{q} is not. As before, we make no notational distinction between an individual element σ ∈ G_{q} and its image in H_{q}. We especially note:

H_{q} =< σ >  ⟺  G_{q} =< σ > or G_{q} =< ⋆σ > .  (10)

Indeed, the orders in G_{q} of σ and ⋆σ are equal up to a possible factor 2, and if they both fell short of |G_{q}|, the image of σ in H_{q} would generate a proper subgroup. Incidentally, since σ^{r} and (⋆σ)^{r} always differ by the factor (⋆)^{r}, we can also say:

ord_{G}(σ) = ord_{G}(⋆σ)  ⟺  ord_{H}(σ) ≡ 0 (mod 2).  (11)

Finally we come to G_{n} ≃ {±1}×G_{q}, whose elements will be denoted by (ε, σ) with ε = ±1 and σ ∈ G_{q}. Then H_{n} equals G_{n} modulo the subgroup consisting of (−1, ⋆) and (1, 1). Again, elements of H_{n} will have the same labels (ε, σ) as their counterparts in G_{n}. Now H_{n} is naturally isomorphic to the subgroup {1} × G_{q} ⊂ G_{n} – since the latter does not contain (−1, ⋆) – and therefore:

H_{n} =< (1, σ) >  ⟺  G_{q} =< σ > .  (12)

We are now ready to return to the question of whether i_{q} = 1 or i_{n} = 1, that is, whether τ : x ↦ x^{2} generates H_{q} or H_{n}, respectively. From (10) and (12) we immediately draw the following conclusion.

Proposition 2:

a) Let q = p^{k} be an odd prime power. Then i_{q} = 1 if and only if (ZZ/qZZ)^{×} is generated by 2 or by −2.

b) Let n = 3q, where q = p^{k} with p > 3. Then i_{n} = 1 if and only if (ZZ/qZZ)^{×} is generated by −2.

Supplementing this, (11) says that (ZZ/qZZ)^{×} is generated by both 2 and −2 if and only if |T_{q}| = |H_{q}| is even, i.e., iff i_{q} = 1 and p ≡ 1 (mod 4). To restate Proposition 2, let us use the notation P(a) to refer to the set of prime powers q such that (ZZ/qZZ)^{×} is generated by the non-zero integer a. Then we have

i_{q} = 1 ⟺ q ∈ P(2) ∪ P(−2) and i_{3q} = 1 ⟺ q ∈ P(−2).  (13)

Moreover, the intersection P(2) ∩ P(−2) consists precisely of those p^{k} ∈ P(2) ∪ P(−2) for which p ≡ 1 (mod 4).

It is well known that p^{k} ∈ P(±2) ⟺ p ∈ P(±2), except in very rare cases. For the sake of completeness, we state this as our final proposition, and recall its proof.

Proposition 3:

Let p be an odd prime such that 2^{p−1} ≢ 1 (mod p^{2}), and let q = p^{k}. Then

i_{p} = 1 implies i_{q} = 1, and i_{3p} = 1 implies i_{3q} = 1.

Proof: For any a ∈ ZZ prime to p, let r_{q}(a) denote the order of a in (ZZ/qZZ)^{×}. Remember that the kernel of the natural surjection (ZZ/qZZ)^{×} ⟶(ZZ/pZZ)^{×} has order p^{k−1}, and that each of its non-trivial subgroups is generated by 1 + p^{j} for j = 1, …, k − 1. It follows that r_{q}(a) = r_{p}(a)p^{k−j(a)}, where j(a) is given by a^{r_{p}(a)} = 1 + bp^{j(a)} or equivalently by a^{p−1} = 1 + cp^{j(a)}, with b and c prime to p. Note that j(−a) = j(a).

Our hypothesis on p now ensures that j(±2) = 1 and hence r_{q}(±2) = r_{p}(±2)p^{k−1}, which, in view of (13), proves what we wanted.

Proposition 3 allows us to check the condition i_{n} = 1 by verifying only what happens in the cases n = p and n = 3p (without involving higher powers of p), provided we stay away from primes for which 2^{p−1} ≡ 1 (mod p^{2}). Such primes are truly exceptional. They are connected with the name of Wieferich, the man who proved that all primes except possibly these fall under the first case of Fermat’s Last Theorem. The only known Wieferich primes are 1093 and 3511, and it has been checked (cf. [6], p.152) that there are no others this side of 3 × 10^{9}. For p = 1093, we have r_{p}(2) = 364 so that i_{p} ≠ 1, and p is of no interest here. However, for p = 3511 it turns out that r_{p}(2) = 1755 and r_{p}(−2) = 3510; therefore both i_{p} and i_{3p} equal 1, but for higher powers of p these indices become non-trivial.

The condition i_{p} = 1, in other words: p ∈ P(2) ∪ P(−2), is satisfied more often than not. The only primes p < 100 which violate it are p = 17, 31, 41, 43, 73, 89, 93. The 10000-th prime, p = 104759, also happens to be the 5632-nd prime in P(2) ∪ P(−2). This suggests a density of over 56% for such primes. In fact, its theoretical value is 1.5 δ ≈ 0.561, where

δ = ∏_{l} (1 − \frac{1}{l^{2} − l} ) ≈ 0.374  (14)

is an infinite product over all primes l. The number δ is the density of primes lying in P(a) for both a = 2 and a = −2, cf. [7], p.xxxvii. Since the overlap of these two sets encompasses exactly half of each, namely the primes ≡ 1 (mod 4), their union has 1.5 times the size of each part.

The tacitly assumed independence of the conditions “p ≡ 1 (mod 4)” and “p ∈ P(a)” can be justified by the same reasons which underlie the derivation of (14), namely the linear independence of the fields Q(ζ_{l}, ^{l} \sqrt{a}). For our purposes, we would just add Q(i) to this list of fields. The density formula (14) was originally conjectured by Emil Artin, cf. [1], pp. viii-ix. The analytic puzzle at the heart of it was first solved by Hooley in 1967, and freed from reliance on a Riemann Hypothesis by Murty [5]. We gratefully acknowledge our debt to Rajiv Gupta for some basic pointers on this topic, which, however, is not central enough to the present context to be discussed in more detail. By way of corroborating (14), suffice it to mention that of the first 10000 primes, 3750 are in P(2), 3752 in P(−2), and 1870 in the intersection.

References.

[1] E. Artin, Collected Papers, Addison-Wesley, New York (1965).

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

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

[4] K. Hoechsmann, Constructing units in commutative group rings, Manuscripta Math. (to appear).

[5] M.R. Murty, An analogue of Artin’s Conjecture for abelian extensions, J. Number Th. 18 (1984) 241 – 248.

[6] P. Ribenboim, 13 Lectures on Fermat’s Last Theorem, Springer, New York (1979).

[7] A.E. Western, J.C.P. Miller, Royal Society Mathematical Tables 9, University Press, Cambridge (1968).