On the Bass-Milnor Index of Abelian p-Groups

by Klaus Hoechsmann

0. Introduction. According to a theorem of Bass and Milnor (cf. [1], Theorem 2), the unit group U(ZA) in the integral goup ring ZA of a finite abelian group A is generated up to finite index by the subgroups UZC, as C ranges over all cycles in A. Actually their theorem concerns K_{1}(ZA) for any finite group A, not necessarily abelian. For the abelian case (and U instead of K_{1}), a simplified version of Bass’s proof is included at the end of this paper, both to serve as reference and to underline that it gives no clue as to the size or nature of the finite index mentioned above.

Letting U̇(ZA) denote the unit group modulo its torsion subgroup, consider the obvious product of inclusions

α_{A} : ∏_{C} U̇(ZC) ⟶ U̇(ZA),

C running over all cyclic subgroups of A. The order ind(α_{A}) of its cokernel will be called the Bass-Milnor index of A. The aim of this paper is to prove the following assertion for any natural prime p.

THEOREM: If the order of A is a p-power, then so is ind(α_{A}).

In view of the results of [6], this entails the strange conclusion that for odd regular p the map α_{A} is surjective. In the case of an elementary abelian A, both the theorem and this consequence have been proved previously (cf. [4] and [5]), and the proof given below is a rather convoluted variation on [4]. It does not use the finiteness established by Bass-Milnor, but depends instead on an “elementary” lemma, whose terms will now be explained.

Fix a prime power p^{m} > 1 and a non- trivial multiplicative character χ : U(R_{m}) → U(C) from the units of the ring R_{m} = Z/p^{m}Z to complex numbers. Define the level l(χ) of χ to be the smallest integer k such that χ(1 + p^{k}Z) = 1, i.e. such that χ can be pulled back from a character on U(R_{k}). The level of χ is a measure of its non-triviality, but the relation between it and the (multiplicative) order of χ is somewhat messy, especially for p = 2. To smoothen the terminology, a level will also be assigned to elements x of any R_{m}-module X: l(x) is the smallest k such that (1 + p^{k}Z)x = x, i.e., it is just the p-logarithm of the (additive) order of x. In particular, for 0 ≠ r ∈ R, the quotient r/p^{m−l(r)} is a unit in R_{l(r)}. Thus it makes sense to by define a function u_{χ} : R_{m} → C by setting

u_{χ}(r) = χ(r/p^{m−l(r)})

if l(r) = l(χ), and 0 otherwise.

Now choose another integer n ≥ 1, and consider the module X of (n+1)-tuples x = (x_{0}, …, x_{n}) with x_{i} ∈ R_{m}. For every cyclic subgroup C ⊂ X pick a generator, and collect these in a set P. Writing P_{χ} = {x ∈ P |l(x) ≥ l(χ)}, form a P_{χ} × P_{χ}-matrix M_{P,χ} by putting

M_{P,χ}(x, y) = u_{χ}(x · y),

where x · y = ∑_{i=0}^{n} x_{i}y_{i} is the usual “dot-product” of the (n + 1)-tuples x, y ∈ P. Note: replacing any element x ∈ P by an equivalent generator vx, with v ∈ U(R_{m}), has the effect of multiplying the x-th row and column of M_{P,χ} by the unimodular complex number χ(v); hence it does not affect

D(χ, n) = | det M_{P,χ}|^{2},

which clearly is an algebraic integer depending only on χ and n. Here, then, comes the crux of our theorem.

MAIN LEMMA: For every non-trivial χ and every n, D(χ, n) is a p-power depending only on l(χ) and n.

After some easy preliminaries, paragraph 2 deduces the theorem from this lemma, the proof of which occupies paragraphs 3 to 6.

Added in proof: Actually the theorem is proved in a more general form (cf. Proposition 1.2) which can be applied, for instance, to certain subfunctors of U̇. The lemma, in turn, is derived from a detailed eigenvalue calculation for the product of M_{P,χ} with its complex conjugate.

1. Generalities. Keeping p^{m} fixed, let ζ_{k} stand for the primitive p^{k}-th root of unity ζ_{k} = exp(2πip^{−k}) for k = 0, 1, …, m. Let C_{p^{m}} denote the full subcategory of finite dimensional Qalgebras whose simple components are isomorphic to some of the cyclotomic fields Q[ζ_{k}]. Note that G = U(R_{m}) acts naturally on all objects of C_{p^{m}}, and that Q[ζ_{k}] = Q[ζ_{l}]^{G_{k}}, where k ≤ l and G_{k} = (1 + p^{k}Z) ∩ G. We shall soon have occasion to refer to the inclusions ι_{k,l} : Q[ζ_{k}] → Q[ζ_{l}] by name. By a lattice we shall mean a finitely generated Z-module.

DEFINITION 1.1: A lattice-valued functor V on C_{p^{m}} is said to be admissible, if

(i) V (Q) = {0}, and V commutes with direct products; (ii) V (ι_{k,l}) bijects V (Q[ζ_{k}]) onto V (Q[ζ_{l}])^{G_{k}} for all 0 ≤ k ≤ l ≤ m.

The second property ensures in particular that the image-lattices never have any (non-trivial) Ginvariant elements, no non-zero submodules on which G = G_{0} acts via the trivial character. This is related to the proviso in the Main Lemma that χ be non-trivial. In fact, the next paragraph will be devoted to deducing the following assertion from that Lemma.

PROPOSITION 1.2: Let A be a finitely generated R_{m}-module and V be an admissible functor on C_{p^{m}}. Then the cokernel of the map

α_{V,A} : ∏_{C} V (QC) ⟶ V (QA),

with C running over all cyclic subgroups of A, is a finite p- group.

Because of property (i), an admissible functor preserves surjections. In particular, any epimorphism A → B of finite groups leads to a surjection coker(α_{V,A}) → coker(α_{V,B}). This has a noteworthy consequence.

REMARK 1.3: It suffices to prove Proposition 1.2 for free R_{m}- modules A.

Trusting that this will be accomplished in due course, the Proposition must still be connected to the Theorem; i.e. U̇(Z−) must be related to V (Q−) for some admissible functor V. This can be done as follows.

Let M(QA) denote the integral closure of ZA in QA, and W (QA) be its unit-group. Under the Wedderburn decomposition QA ≃ ∏ Q[ζ_{k}], the ring M(QA) corresponds to ∏ Z[ζ_{k}]. Hence W (QA) ≃ ∏ U(Z[ζ_{k}]) is well on its way to being an admissible functor (just shed the torsion …), but unfortunately it is too big. To cut it down to size, we take only the subgroup W_{1}(QA) of 1units, i.e. those which are congruent to 1 modulo the ideal ΔM(QA) generated by the differences {a − 1 | a ∈ A}. It is important to note that this is smaller than the kernel of the so-called augmentation map κ : QA → Q (sum of the coefficients) restricted to M(QA); in fact it is the direct product of ideals I_{k} = (ζ_{k} − 1)Z[ζ_{k}]. Of course, in ZA the ideal Δ(ZA) generated by the same differences is the kernel of κ, and hence U(ZA) = {±1} × U_{1}(ZA), where the subscript 1 again means ≡ 1 modulo Δ(ZA).

Thus, after factoring out the torsion everywhere, U̇(ZA) = U̇_{1}(ZA) can be seen as a subgadget of Ẇ_{1}(QA), and the latter finally is the desired V (QA). For fields this means that V (Q[ζ_{k}]) is the group U(Z[ζ_{k}]) ∩ (1 + I_{k}) taken modulo its torsion subgroup. Property (ii) holds by Galois theory and the fact that I_{k} = I_{l} ∩ Q[ζ_{k}] for k ≤ l. The gap between Proposition 1.2 and the Theorem is now bridged by the next little lemma.

LEMMA 1.4: U_{1}(ZA) has p-power index in W_{1}(QA).

Proof: Let u ∈ W_{1}(QA) ⊂ W_{1}(Q_{p}A). By general principles (cf.[7], proof of II.2.9,pp.49-50), U_{1}(Z_{p}A) is of finite index in W_{1}(Q_{p}A), but since the latter is a Z_{p}- module (cf. [3],II.15.5), this index must be a p-power. Thus v = u^{N} ∈ Z_{p}A for some p-power N. But v ∈ M(QA) ⊂ \frac{1}{|A|}ZA, hence v ∈ ZA.

As a consequence, U̇(ZA) has p-power index in V (QA), as does the composite

∏_{C} U̇(ZC) ⟶ ∏_{C} V (QC) ⟶ V (QA),

by Proposition 1.2. Factorization via U̇(ZA), however, shows this to be divisible by the Bass-Milnor index, completing our bridge.

To change the topic somewhat, let Γ be a finite group and K be a commutative ring in which the order |Γ| is invertible. Then the group ring KΓ contains the obvious idempotents |Γ|^{−1}T and 1−|Γ|^{−1}T, where T = ∑_{g∈Γ} g is the sum over all group elements; these decompose any KΓ-module X into X^{Γ} ⊕ kerT. Applying this recursively to Γ = G_{k−1} and X = Y^{G_{k}}, for some KG- module Y and k = m, …, 2, we obtain the decomposition

Y = Y_{1} ⊕ ⋯ ⊕ Y_{m}, with Y_{k} = Y^{G_{k}} ∩ kerT_{k−1} and Y_{1} = Y^{G_{1}}.

Intuitively, Y_{k} is the part of Y which is fixed under G_{k} but, for k > 1, is moved full circle by the larger group G_{k−1}. The reason for stopping the process at k = 2 , i.e. Γ = G_{1}, is that [G : G_{1}] is prime to p and the K to be chosen later might not allow division by this index. Of course, this means that there is nothing to decompose if m = 1, which is just as well.

Another way to look at the meaning of Y_{k} is contained in the following observation, whose details are left to the reader.

REMARK 1.5: If K = C and if Y = ∏_{χ} Y_{χ} is the decomposition according to characters χ : G → U(C), then Y_{k} = ∏_{l(χ)=k} Y_{χ}.

Since these components are defined in terms of the G-action, which is respected by every morphism in C_{p^{m}}, the next assertion is not surprising either.

LEMMA 1.6: Let p be invertible in the commutative ring K, and let V be an admissible functor. Then there is a functorial decomposition V = V_{1} ⊕ ⋯ ⊕ V_{m}, with V = K ⊗ V, V_{1} = V^{G_{1}}, and V_{k} = V^{G_{k}} ∩ kerT_{k−1} for k > 1. For h ≤ k the maps V_{h}(ι_{k,l}) are isomorphisms.

The proof is immediate from the discussion above. In fact, the statement would hardly be worth making, except to avoid blurring the next paragraph with these notational niceties.

There is a similar motive (applying to paragraph 5) for recording the last of these preparatory lemmas, which concerns a subgroup chain {1} = Γ_{h} ⊂ Γ_{h−1} ⊂ ⋯ ⊂ Γ_{0} = Γ and an element s = ∑_{g∈Γ} c(w(g))g of the ring KΓ whose coefficient for any g ∈ Γ depends only on w(g) = max{i | g ∈ Γ_{i}}, i.e. on the location of g in the chain. Assume that K is a field.

LEMMA 1.7: In the preceding context, let χ : KΓ → K be a homomorphism which is non-trivial on Γ_{h−1}. Then χ(s) = c(h) − c(h − 1).

Proof: Defining c(−1) = 0 and t_{k} = ∑_{w(g)=k} g, for 0 ≤ k ≤ h, we have

s = ∑_{k=0}^{h} c(k)t_{k} = ∑_{k=0}^{h} t_{k} ∑_{j=0}^{k}(c(j) − c(j − 1)) = ∑_{j=0}^{h}(c(j) − c(j − 1)) ∑_{k=j}^{h} t_{k}.

Now, χ(∑_{k=j}^{h} t_{k}) = ∑_{w(g)≤j} χ(g) = ∑_{g∈Γ_{j}} χ(g) is the sum over a non-trivial subgroup of U(K), hence = 0, unless j = h. This finishes the proof.

2. Reduction to Main Lemma. In preparation for the announced reduction, we need to describe the Wedderburn decomposition of QA more concretely than usual, not just “up to isomorphism” but in terms of nuts and bolts. Any functional f : A → R_{m} yields a Q-algebra homomorphism QA → Q[ζ_{l(f)}] by a ↦ ζ_{m}^{f(a)}, and every G-orbit in Hom(A, R_{m}) constitutes a set of Q- isomorphic irreducible representations, with G = U(R_{m}) as before. Letting F ⊂ Hom(A, R_{m}) be a set of representatives of G -orbits, we get an algebra isomorphism:

QA ⟶ ∏_{f∈F} Q[ζ_{l(f)}],

induced by a ↦ (…, ζ_{m}^{f(a)}, …). This is the ”Wedderburn decomposition” with respect to F. In case A = C_{x} is cyclic with a fixed generator x, we can even be more explicit, inducing the map by x ↦ (…, ζ_{k}, …), with 0 ≤ k ≤ l(x). It is crucial to relate these two methods of book-keeping for elements x ∈ A. Indeed, for a single x ∈ A and a single f : A → R_{m}, they fit into a commutative square

\begin{matrix}QC_{x} & ⟶ & ∏_{k≤l(x)} Q[ζ_{k}] \\ ↓ & & ↓ \\ QA & ⟶ & Q[ζ_{l(f)}] \end{matrix}  (1)

with the top arrow representing the Wedderburn decomposition. The point is to describe the right vertical arrow, and here is what it does:

1. Project ∏_{k≤l(x)} Q[ζ_{k}] → Q[ζ_{l′}] onto the l′-th component, where l′ = l(f(x)).

2. Apply the automorphism of Q[ζ_{l′}] which changes ζ_{l′} ↦ ζ_{m}^{f(x)}

3. Include Q[ζ_{l′}] → Q[ζ_{l(f)}] using ι_{l′,l(f)}.

The stage is now clear for deducing Proposition 1.2 from the Main Lemma. This will be achieved by an ascending chain of “claims”, each of which will imply the previous one. To lighten the notation, we agree to omit reference to Q from our functor V, writing simply V (A), V (ζ_{k}), etc. Moreover, we take advantage of Remark 1.3, and let A = R_{m}^{n+1} with n ≥ 1. Any f : A → R_{m} can then be obtained in the form y^{∗} : x ↦ x · y, so that the set P of representatives of G-orbits in A (cf. Introduction) can also serve as the set F mentioned at the beginning of this paragraph. All in all, we have a commutative square

\begin{matrix}∏_{x∈P} V (C_{x}) & ⟶ & ∏_{x∈P} ∏_{k≤l(x)} V (ζ_{k}) \\ ↓ & & ↓ \\ V (A) & ⟶ & ∏_{y∈P} V (ζ_{l(y)}) \end{matrix}  (2)

whose horizontal arrows are lattice-isomorphisms and whose left vertical is our dear old α. All these maps are compatible with the G-action. Referring to the right vertical as γ, Proposition 1.2 boils down to Claim 1: The cokernel of γ is a finite p-group.

Tensoring all these lattices by the ring of quotients K = Z[p^{−1}], and abbreviating V = K ⊗ V as well as δ = K ⊗ γ, we must show that δ is surjective. This can be accomplished by looking separately at each component of the decomposition V = V_{1} ⊕ ⋯ ⊕ V_{m} which exists by virtue of Lemma 1.6. So, we are down to Claim 2: The map δ_{h} is surjective, for h = 1, …, m.

Aforesaid map is depicted as the left vertical arrow in our third and final commutative square, whose horizontal arrows are made from the routine identifications V_{h}(ζ_{k}) ≃ V_{h}(ζ_{m}) via ι_{k,m} compounded, in the case of the top arrow, by a drastic thinning out of the Wedderburn components belonging to each C_{x}: of all the components corresponding to indices k with h ≤ k ≤ l(x) only the one belonging to k = h is kept, the others are mapped to zero. Here is the diagram:

\begin{matrix}∏_{x∈P_{h}} ∏_{h≤k≤l(x)} V_{h}(ζ_{k}) & ⟶ & ∏_{x∈P_{h}} V_{h}(ζ_{m}) \\ ↓ & & ↓ \\ ∏_{y∈P_{h}} V_{h}(ζ_{l(y)}) & ⟶ & ∏_{y∈P_{h}} V_{h}(ζ_{m}) \end{matrix}  (3)

The bottom arrow represents a straightforward isomorphism and the top one is a projection onto a direct summand. Therefore, letting ϵ_{h} stand for the morphism on the right of this square, it would certainly suffice to prove Claim 3: The map ϵ_{h} is an isomorphism, for h = 1, …, m.

For fixed x, y the map δ_{h} is still described fairly well by items 1. to 3. following diagram (1). Translating from that description, here is what happens under the y-th component of our ϵ_{h} to the x-th component of the product ∏_{x} V_{h}(ζ_{m}): 1’. Annihilate if l′ = l(x · y) does not equal h. 2’. Otherwise apply the automorphism ζ_{h} ↦ ζ_{m}^{x·y}.

Note that the third stage is now subsumed in the ι_{k,m} of the bottom arrow. The main advantage of our new arrangement is that the map ϵ_{h} can be described as a matrix M_{h} with entries in the ring-endomorphisms of Q[ζ_{h}], namely M_{h}(x, y) = u_{h}(x · y) where, for r ∈ R_{m},

u_{h}(r) = \{ \begin{matrix}r/p^{m−h}, & if l(r) = h; \\ 0, & otherwise. \end{matrix}

As written, these entries are 0 or in the group U(R_{h}), so they act on X_{h} = V_{h}(ζ_{m}) as elements of the group ring K_{h} of that group over Z[p^{−1}]. Therefore the matrix M_{h} defines an isomorphism, if and only if Claim 4: d_{h} = det M_{h} ∈ K_{h} acts on X_{h} as an isomorphism.

To show this, it would obviously suffice to find that d_{h}c acts invertibly on X_{h} for a suitable c ∈ K_{h}. Indeed, let the matrix M_{h}^{−} be defined by M_{h}^{−}(x, y) = M_{h}(x, y)^{−1} wherever that makes sense (and 0 elsewhere), and let d_{h}^{−} be its determinant. Since p is a unit in K_{h}, it is enough to establish Claim 5: d_{h}d_{h}^{−} acts on X_{h} as a p-power.

Now we are very close to the Main Lemma. In fact, a character χ : U(R_{h}) → U(C) yields a ring homomorphism χ : K_{h} → C, and if χ is of level h the definitions given here obviously match those of the Introduction, so that χ(d_{h}d_{h}^{−}) = D(χ, n). But by Remark 1.5 all characters occurring in the C[G]- module C ⊗ X_{h} are of level h.

3. Hermitian Product. Just as every divisor of p^{m} occurs both as the order of a subgroup and of a factor group of R_{m}, the quantity m − l(x) inevitably crops up alongside the level l(x) of any element x of an R_{m}-module. Instead of complicating several formulas with too many minus signs, let us agree to use the notation v(x) = m − l(x). For the character χ : G → U(C) of the Main Lemma, let us set the level l(χ) = h and put m − h = κ. Note that l(x) ≥ l(χ) is exactly the same condition as v(x) ≤ κ.

Since the basic set-up P ⊂ A = R_{m}^{n+1}, as described in the Introduction, will be kept fixed, it need not always be referred to by the notation for the matrix M_{χ} whose determinant is the subject of the Main Lemma. Actually, that lemma is about the determinant of the Hermitian matrix S_{χ} = M_{χ}Mχ̄ whose entries are given by

S_{χ}(x, y) = ∑_{z∈P} u_{χ}(x · z)ū_{χ}(z · y),

for x, y ∈ P_{χ}, i.e. v(x), v(y) ≤ κ. Note that the restriction l(z) ≥ l(χ) is automatically enforced by the function u_{χ}. Still, the first problem with this formula is the unknown and possibly chaotic nature of the summation index z ∈ P. This is easily straightened out: imagine |G| such formulas, each obtained from the one displayed by writing gz instead of z everywhere on the right (g ∈ G), and then letting the χ(g) be cancelled by the χ̄(g) in each summand; finally add together all these equations. The index of the resulting sum will now run over complete G-orbits, covering each non-trivial orbit Gz exactly p^{v(z)} times, as this is the order of the fix-group G_{l(z)} = {g| gz = z} of any z ≠ 0. Here is the result of this symmetrizing summation:

|G|S_{χ}(x, y) = ∑_{z∈A} p^{v(z)}u_{χ}(x · z)ū_{χ}(z · y).  (4)

CONVENTION 3.1: In the subsequent formulas, it will be supposed that l(x) ≤ l(y), in other words that v(y) ≤ v(x). Since S_{χ} is Hermitian, this is a harmless constraint.

The relationship between the indices y and x can be made more visible by transforming x, y to a kind of standard form by a matrix L ∈ GL(n + 1, R_{m}). We picture a 2 × (n + 1)-matrix in which y forms the first, x the second row. Applying elementary column operations, we can reduce all but one of the entries in the first row to zero, and so on. The outcome of this exercise is

\begin{aligned}yL & = & [ & p^{μ}, & 0, & …, & 0] \\ xL & = & [ & r, & p^{λ}, & …, & 0] \end{aligned}  (5)

where the off-diagonal term r can be made zero unless v(r) < λ, in which case v(r) = v(x) ≥ μ. In all instances we can therefore write r = tp^{μ}, with t ∈ R_{m} determined modulo p^{λ−μ}. Incidentally, λ ≥ μ, because λ < μ would imply v(x) = λ and v(x) < μ = v(y), contrary to our convention.

The next step toward understanding the sum S_{χ}(x, y) comes from the observation that (4) is not affected if we substitute zN for z in each summand, where N denotes a fixed element of GL(n + 1, R_{m}), say the transpose of the L used above. The individual summands now appear as

p^{v(z)}u_{χ}(z_{0}tp^{μ} + z_{1}p^{λ})ū_{χ}(z_{0}p^{μ}),

where z = [z_{0}, …, z_{n}]. The second factor in this is ≠ 0 if and only if l(z_{0}p^{μ}) = l(χ), i.e. iff v(z_{0}) = κ − μ, i.e. iff z_{0} = gp^{κ−μ} with 0 ≤ κ − μ < m and g ∈ G. Since the fix-group of such a z_{0} has order p^{κ−μ}, the number of possible values for z_{0} is p^{μ−κ}|G|. Therefore, if we take only one of these, that is, if we fix z_{0} = p^{κ−μ} and restrict the summation to z′ = [z_{1}, …, z_{n}] ∈ A′ = R_{m}^{n}, formula (4) mutates to

p^{κ−μ}S_{χ}(x, y) = ∑_{z′∈A′} p^{(κ−μ)∩v(z′)}u_{χ}(tp^{κ} + z_{1}p^{λ}),  (6)

where the notation a ∩ b is used instead of the lengthier min{a, b}.

DEFINITION 3.2: A pair x, y ∈ X from a free R_{m}-module X is k-congruent if v(x − y) ≥ k; is k-similar if gx, y is k-congruent for some g ∈ G; and is k-dependent if tx, y or x, ty is k-congruent for some t ∈ R_{m}. It will be called quasi-congruent (quasi-similar) if it is k-congruent (k-similar) for some k > v(x), v(y).

LEMMA 3.3: With v(y) ≤ v(x), let λ, μ, t be the parameters appearing in (5) and (6). Then a) μ = v(y), and λ is the largest integer such that x, y is λ-dependent.

b) x, sy is λ-congruent if and only if s ≡ t modulo p^{λ−μ}.

c) x, y is quasi-similar if and only if μ < λ and t is a unit.

Proof: (a) is clear from (5); (b) amounts to the observation that v((s−t)y) ≥ λ iff v(s−t) ≥ λ−μ. The if-part of (c) is clear; the converse follows from (b).

Having thus obtained an intrinsic characterization of the terms of our formula for S_{χ}(x, y), we turn back to it with a view to ”divide” the expression tp^{κ} + z_{1}p^{λ} by p^{κ}. Since v(z_{1}p^{λ}) < κ implies that u_{χ}(tp^{κ} + z_{1}p^{λ}) = 0, we might as well constrain z_{1} by the condition v(z_{1}) + λ ≥ κ, that is z_{1}p^{λ} = sp^{κ} for suitable s ∈ R_{m}. Actually, as κ + h = m, this element s is only determined modulo p^{h}. It is systematically obtained from z_{1} by first applying the surjection R_{m} → p^{λ}R_{m} ( multiplication) and then the isomorphism p^{κ}R_{m} → R_{h} ( division). What happens here is best understood by visualizing R_{m} as the set of m-digit p-adic integers, and the maps as a λ -fold forward shift (overflow allowed but lost) and a κ-fold backward shift (no underflow allowed), respectively. The composite of these maps will be denoted by ϕ_{κ}^{λ}(z_{1}) and is defined only if v(z_{1}) ≥ κ − λ (no condition if κ < λ). Similarly, first multiplying and then dividing the parameter t ∈ R_{m} by p^{κ} we get a certain element t_{h} = ϕ_{κ}^{κ}(t) ∈ R_{h}, which is none other than the image of t under the canonical map R_{m} → R_{h}, and like t is unique modulo p^{λ−μ}. The formula now looks like:

p^{κ−μ}S_{χ}(x, y) = ∑_{z′∈A′,v(z_{1})≥κ−λ} p^{(κ−μ)∩v(z′)}χ(t_{h} + ϕ_{κ}^{λ}(z_{1})),  (7)

with the understanding that χ : R_{h} → C takes non-units to zero.

We now leave aside S_{χ} for a while, in order to make an adjustment on the index set P.

DEFINITION 3.4: A subset F ⊂ A is fully transversal if k-similarity implies k-congruence for any pair x, y from F and any k = 1, …, m.

LEMMA 3.5: The set of P ⊂ A of G-representatives may be chosen to be fully transversal.

Proof: For every x = [x_{0}, …, x_{n}] ∈ A, let i(x) = min{i | v(x_{i}) = v(x)}. Take P be the set of (n + 1)- tuples x ≠ 0 whose i(x)-th component is a power of p (possibly 1). Clearly this catches exactly one element in every non-trivial G-orbit.

Now take x, y ∈ P with v(y) ≤ v(x). From v(x − gy) ≥ k we must deduce that v(x − y) ≥ k. To make the task non-trivial, we may assume that v(y) < k. Then v(gy) = v(y) < v(x) would imply v(gy −x) = v(gy) < k, and not ≥ k; hence we must have v(x) = v(y). Moreover, i(x) ≤ i(gy) or else the difference would not get shorter at the i(x)-th component. Hence i(x) = i(gy) = i(y) by symmetry.

Now let i be that common index i(x) = i(y). By definition of P, we know that x_{i} = y_{i} = p^{μ} where μ = v(x) = v(y) < k. Since p^{μ} − gp^{μ} ∈ p^{k}R_{m}, it follows that 1 − g ∈ p^{k−μ}R_{m}. Hence x ≡ gy ≡ y modulo p^{k}, as was to be shown.

To sum up the results of this paragraph, the following lemma can now be stated without further proof.

SUMMARY 3.6: Let P ⊂ A be a fully transversal set of G- representatives, and let x, y ∈ P_{χ} with v(x) ≥ v(y) = μ, and λ be the largest integer such that x, y is λ-dependent. Then there is a t_{h} ∈ R_{h} such that (7) holds; with (i) μ < λ and t_{h} = 1 if x, y is quasi-similar (hence quasi-congruent), and (ii) t_{h} ≡ 0 (mod p) if not.

It is worth noting that t_{h} is unique if λ − μ ≥ h. At the other extreme, if λ = μ we can take t_{h} = 0; so this falls under case (ii). For 0 < λ − μ < h, there is some vagueness (mod p^{λ−μ}) to the value of t_{h}, but its status as unit or non-unit is fixed; if it is a unit, we can and will put t_{h} = 1.

4. Block Decomposition. In this short paragraph, S_{χ}(x, y) will be computed in two extreme situations, namely x, y not being quasi-similar and x, y being equal, and a certain organizational matter will be attended to. We keep the notation and terminology of the last paragaph, but need not yet assume that P is fully transversal.

LEMMA 4.1: If x, y ∈ P_{χ} are not quasi-similar, then S_{χ}(x, y) = 0. Proof: Since S_{χ} is Hermitian, we may assume v(x) ≥ v(y) = μ and apply formula (7) with t_{h} ≡ 0 (mod p). Then χ(t_{h}+ϕ_{κ}^{λ}(z_{1})) = 0 unless ϕ_{κ}^{λ}(z_{1})) is a unit, which happens if and only if v(p^{λ}z_{1}) = κ, i.e. v(z_{1}) = κ − λ (remember that κ < m). But there are no such z_{1}, and hence everything is zero, unless κ ≥ λ, in which case the homomorphism ϕ_{κ}^{λ} : B = {b ∈ R_{m}| v(b) ≥ κ − λ} → p^{κ}R_{m} → R_{h} is surjective with kernel of order p^{λ}, so that every w ∈ U(R_{h}) occurs exacty p^{λ} times in the image of ϕ_{κ}^{λ}. Letting z′′ = [z_{2}, …, z_{n}] run over A′′ = R_{m}^{n−1} (possibly the trivial group) , we get

\begin{aligned}p^{κ−μ}S_{χ}(x, y) & = ∑_{z′′} ∑_{b∈B} p^{(κ−μ)∩v(b)∩v(z′′)}χ(t_{h} + ϕ_{κ}^{λ}(z_{1})) \\ & = ∑_{z′′} p^{(κ−λ)∩v(z′′)+λ} ∑_{w∈W} χ(t_{h} + w), \end{aligned}

with W = U(R_{h}). The first line is just a literal translation of (7), the second uses the fact that v(b) = κ − λ ≤ κ − μ, etc. Since w ↦ w + t_{h} yields a permutation of W, the term t_{h} can be omitted from the second line, and since W is non-trivial the final sum is zero (remember that h ≥ 1), as was to be shown.

For any subset Q ⊂ P_{χ} let (S_{χ}|Q) denote the Q×Q-matrix obtained by restriction to x, y ∈ Q. Since P_{χ} is the disjoint union of quasi-similarity classes, the preceding lemma can also be stated as follows.

COROLLARY 4.2: S_{χ} is the direct sum of (S_{χ}|Q) as Q ranges over quasi-similarity classes in P_{χ}.

As the computation of the diagonal entries does not fit well into the context of the next paragraph, it will be used to fill out the present one. The following notation will be used repeatedly.

NOTATION 4.3: If T is an indeterminate, define the polynomials π_{k}(T) by

π_{k}(T) = \{ \begin{matrix}T^{m}, & if k = m; \\ (p^{m−k} − p^{m−k−1})T^{k}, & if k < m. \end{matrix}

LEMMA 4.4: For v(x) = μ ≤ κ we have

p^{κ−μ}S_{χ}(x, x) = ∑_{k=0}^{κ−μ} π_{m−k}(p^{n}).

Proof: Let us put q = p^{n}. Then q^{m−k} is the number of z′ ∈ A′ such that v(z′) ≥ k, for 0 ≤ k ≤ m. Writing κ − μ = s, we have by (7):

p^{s}S_{χ}(x, x) = ∑_{z′∈A′} p^{s∩v(z′)} = q^{m−s}p^{s} + ∑_{v(z′)<s} p^{v(z′)} = q^{m−s}p^{s} + ∑_{0≤k<s}(q^{m−k} − q^{m−k−1})p^{k},

whence the result by rearrangement of summands (summation by parts).

If m = 1, the two extreme cases treated here are the only ones possible, and the matrix S_{χ} is diagonal. Since κ < m, we must have κ = μ = 0, and the diagonal terms are S_{χ}(x, x) = q = p^{n} in happy agreement with the Lemma of [4].

5. Positive Entries. Before getting into the actual agenda of this paragraph we will present its crucial computation in abstract form, out here where the plot is not so thick.

LEMMA 5.1: Let B be cyclic of order p^{l}, and ϕ : B → R_{h} be a homomorphism whose kernel is of order p^{λ}, with 0 < λ < l. Let f : {0, …, l} → C be a function which is constant up to ν > 0, i.e. such that f(k) = f(ν) for k ≤ ν. If χ : U(R_{h}) → C is a character of level h, we have

∑_{b∈B} f(l(b))χ(1 + ϕ(b)) = ∑_{ν≤k≤λ} p^{k}(f(k) − f(k + 1)),

it being understood that χ takes non-units to zero, and that the second sum is trivial if λ < ν. Proof: The sum to be computed is obviously equal to ∑_{a∈D} F (a)χ(1 + a), where D = {a ∈ R_{h}| 1 + a ∉ pR_{h}} and F (a) = ∑_{ϕ(b)=a} f(l(b)). We claim that

F (a) = \{ \begin{matrix}p^{λ}f(λ + 1) + ∑_{ν≤k≤λ} p^{k}(f(k) − f(k + 1)), & if a = 0; \\ p^{λ}f(l(a) + λ), & if 1 ≤ l(a) ≤ l − λ; \\ 0, & if l − λ < l(a). \end{matrix}  (8)

First, let a ≠ 0. Then ϕ(b) = a means l(b) = l(a) + λ; moreover there are exactly p^{λ} such b if l(a) + λ ≤ l and none otherwise — whence the last two cases of the formula. In particular, F (a) = p^{λ}(1 + λ) for l(a) = 1, for instance a = p^{h−1}.

Next, F (0) = ∑_{l(b)≤λ} f(l(b)) is easy if λ ≤ ν, since then all summands equal f(ν) = f(λ), and their number is again p^{λ}. This agrees with the first line of (8), where the second term is trivial for λ < ν.

Finally, suppose that ν < λ. Then

F (0) = ∑_{l(b)<ν} f(ν) + ∑_{ν≤l(b)≤λ} f(l(b)) = p^{ν−1}f(ν) + ∑_{ν≤j≤λ}(p^{j} − p^{j+1})f(j),

which gives (8) after summation by parts.

To finish the proof we call on Lemma 1.7, which we apply to Γ = U(R_{h}) with the usual filtration of subgroups Γ_{k} = {1 + p^{k}R_{h}} ∩ U(R_{h}). Since F (a) depends only on l(a), this lemma says that the sum to be computed equals F (0) − F (p^{h−1}) = F (0) − p^{λ}f(λ + 1). This easily yields the desired conclusion.

LEMMA 5.2: Let 0 < ν ≤ m be integers, and d : {0, …, m} → Z_{≥0} be a function which is constant up to ν and such that d(k + 1) = d(k) − 1 if ν ≤ k < m. Let I be a finite set, and e : I → Z_{≥0} be any function. Putting

f(k) = ∑_{i∈I} p^{e(i)∩d(k)}  we have  f(k) − f(k + 1) = \{ \begin{matrix}0, & for 0 ≤ k < ν; \\ N_{k}(p^{d(k)} − p^{d(k+1)}), & if ν ≤ k < m, \end{matrix}

where N_{k} denotes the number of i ∈ I such that e(i) ≥ d(k). Proof: Obviouly f(k) is constant for k ≤ ν, because d(k) is. For the rest it suffices to subtract the identities

\begin{aligned}f(k) & = ∑_{e(i)<d(k)} p^{e(i)} + ∑_{e(i)≥d(k)} p^{d(k)}, \\ f(k + 1) & = ∑_{e(i)≤d(k+1)} p^{e(i)} + ∑_{e(i)>d(k+1)} p^{d(k)}, \end{aligned}

whose summation indices match because, for ν ≤ k < m, the condition < d(k) is the same as ≤ d(k + 1), as d(k) − d(k + 1) = 1.

We now turn back to the context and notation of the previous two paragraphs.

PROPOSITION 5.3: Let x, y ∈ P_{χ} be quasi-congruent, with v(x) = v(y) = μ, and let h = l(χ). Then

p^{m}S_{χ}(x, y) = p^{h+μ} ∑_{h+μ≤k≤λ} π_{k}(p^{n}),

where λ is the largest integer ≤ m for which x, y are λ-congruent. Proof: Let us first agree that the case λ = m, i.e. x = y, has been dealt with in Lemma 4.4. To recognize the two results as identical, we have to recall the relation κ − μ = m − (h + μ) and perform the change of variable k ↦ m − k. Henceforth we may assume that μ < λ < m.

Setting B = {b ∈ R_{m}| v(b) ≥ κ − λ}, we apply formula (7) with t_{h} = 1 to get

p^{κ−μ}S_{χ}(x, y) = ∑_{b∈B} s(b)χ(1 + ϕ_{κ}^{λ}(b))  with  s(b) = ∑_{z′′∈A′′} p^{v(z′′)∩v(b)∩(κ−μ)},

just as in the proof of Lemma 4.1. This can be written as s(b) = f(l(b)), where f has the shape shown in Lemma 5.2, with I = A′′, e = v, and d(k) = (m − k) ∩ (κ − μ). The set-up now obviously calls for the application of Lemmas 5.1 and 5.2, but we must first check out their parameters and premises.

B is either all of R_{m} (if κ − λ < 0) or p^{κ−λ}R_{m}, hence its order is p^{l}, with l = m ∩ (h + λ). Since h ≥ 1, we have λ < h + λ; together with the assumption that λ < m, this assures 0 < λ < l, as is required by the first sentence of 5.1. The role of ν will be played by h + μ, which is positive because h is. The function f needs to be defined only on {0, …, l}, so the possibly larger domain of definition of d is ample. Constancy up to ν is inherited from d, which is stationary until (m − k) reaches (κ − μ), i.e. k = m − κ + μ = ν. All the premises of 5.1 and 5.2 are now verified.

We are almost ready to pull the switch. The d(k) appearing in the conclusion of 5.2 is just m − k for the range in question, but what is N_{k}? It is the number of z′′ ∈ A′′ = R_{m}^{n−1} for which v(z′′) ≥ m − k, namely (p^{n−1})^{k}. Altogether we get

∑_{h+μ≤k≤λ} p^{k}(p^{n−1})^{k}(p^{m−k} − p^{m−k−1}),

as was to be shown.

The final result depends only on h = l(χ) (fixed throughout), μ = v(x) = v(y) (fixed within each quasi-similarity class), and λ which can range from μ + 1 to m. It is 0 for λ less than μ + h and takes on strictly increasing integer values beyond that point. Its triviality for small λ leads to a refinement of the block decomposition of 4.2: for S_{χ}(x, y) to be non-zero, it is necessary (and sufficient) that x, y be (h + μ)-congruent; let us call them h-strictly congruent if v(x) = v(y) = μ, and v(x − y) ≥ h + μ.

COROLLARY 5.4: If P is fully transversal, S_{χ} is the direct sum of (S_{χ}|Q) as Q ranges over l(χ)-strict congruence classes of P_{χ}.

The full transversality (cf.3.4) is necessary to force quasi-similar pairs (cf. 4.2) to be quasicongruent, so that 5.3 can be applied.

6. Eigenvalues. Again we begin by presenting the main computations in abstracto. Let q ≥ 2, s ≥ 0 be integers, and a_{0}, …, a_{s} be a finite sequence of elements of some field F. The q^{s} × q^{s}- matrix L_{q}(a_{0}, …, a_{s}) is defined recursively as follows. L_{q}(a_{0}) = [a_{0}], and

L_{q}(a_{0}, …, a_{s}) = a_{s}I_{q^{s}} + E_{q} ⊗ L_{q}(a_{0}, …, a_{s−1}),

where E_{q} is the q × q-matrix with all entries = 1, and ⊗ denotes the Kronecker product (cf. [2], p.69). A more visual description would be: take L_{q}(a_{0}, …, a_{s−1}), blow up each entry to a q × q-block, and add a_{s} to the main diagonal of the result. REMARK 6.1: We need to formalize this description somewhat. For every natural number 0 < t ≤ s, let ρ : Q_{t} → Q_{t−1} be a q-to -1 surjection of finite sets, and let Q_{0} have cardinality 1. All these maps get the same name ρ, as there is no danger of confusion. (For instance, ρ might be the dropping of the last digit, if Q_{s} is the set of s-digit q-adic numerals.) Such a structure clearly amounts to the same as the imposition on Q_{s} of a nested sequence of equivalence relations {∼_{k} | k = 0, …, s} such that each class defined by ∼_{k} breaks up into exactly q classes given by ∼_{k+1}.

If we imagine our matrix as a function on the Cartesian product Q_{s} ×Q_{s}, the recursion process consists of pulling back along ρ × ρ and then adding a_{s} to the diagonal. A simple induction now proves that the i, j-th entry of L_{q}(a_{0}, …, a_{s}) equals a_{0} + ⋯ + a_{s−k}, where k is the smallest integer such that ρ^{k}(i) = ρ^{k}(j); or in the other language, a_{0} + ⋯ + a_{t} ,where t is the largest integer such that i ∼_{t} j.

LEMMA 6.2: The eigenvalues of L_{q}(a_{0}, …, a_{s}) are ϵ_{k} = a_{s}+qa_{s−1}+⋯+q^{k}a_{s−k}, with k = 0, …, s. Their respective multiplicities are m_{s} = 1 and m_{k} = q^{s−k} − q^{s−k−1}, for k < s.

Proof: Clear for s = 0. For s > 0, we have

E(L_{q}(a_{0}, …, a_{s})) = a_{s} + E(E_{q} ⊗ L_{q}(a_{0}, …, a_{s−1})) = a_{s} + E(E_{q}) · E(L_{q}(a_{0}, …, a_{s−1})),

where E denotes the generic eigenvalue. Now, E_{q} has eigenvalues 0 (multiplicity q − 1) and q (multiplicity 1). Thus, for L_{q}(a_{0}, …, a_{s}) we get a_{s} with multiplicity (q − 1)q^{s−1}, and a_{s} + qE(L_{q}(a_{0}, …, a_{s−1})) with the same multiplicity as before — whence the result.

COROLLARY 6.3: Let b_{0}, …, b_{s} be a finite sequence in F, and put a_{t} = q^{t}(b_{t} − b_{t+1}) for t < s, setting a_{s} = q^{s}b_{s}. Then the eigenvalues of L_{q}(a_{0}, …, a_{s}) are ϵ_{k} = q^{s}b_{s−k} with multiplicity m_{k} as in 6.2. In particular, if b_{t} = p^{s−t}, we get ϵ_{k} = q^{s}p^{k}.

The point of this corollary, whose proof is a trivial verification, is to narrow the gap between the matrices considered here and those of the last paragraph. In fact, with q = p^{n} and s = m − ν = m − (h + μ) = κ − μ as in the proof of 5.3, we shall presently see that

p^{s}(S_{χ}|Q) = L_{q}(π_{ν}(q), …, π_{ν+s}(q)),  (9)

where Q ⊂ P_{χ} is an h-strict congruence class.This will lead, via 6.3, to the following conclusion.

PROPOSITION 6.4: In the situation of 5.4, each submatrix (S_{χ}|Q) has eigenvalues σ_{k} = q^{m}p^{k−s} with multiplicities m_{k} as in 6.2, for 0 ≤ k ≤ s = κ − ν. Proof: A typical Q looks like the set of all n + 1-tuples of the form [p^{μ}, x_{1}, …, x_{n}] with l(x_{i}) ≤ s, and therefore has the right size. If we mimic 6.1 by writing x ∼_{t} y to mean v(x − y) ≥ ν + t, the second part of 6.1 translates 5.3 into (9). On the other hand, if we construct the sequence a_{t} by the prescription of 6.3, setting b_{t} = p^{s−t}, we easily check that a_{t} = q^{−ν}π_{ν+t}(q). By 6.3 and formula (9), the eigenvalues of p^{s}(S_{χ}|Q) are therefore q^{ν}ϵ_{k}, whence the proposition.

Since q = p^{n} with n ≥ 2, all eigenvalues of S_{χ} are integral p-powers — provided that P was chosen fully transversal. This proves the Main Lemma.

7. Appendix: Finiteness A Priori. The following is a proof, in the abelian case, of a general theorem by Bass and Milnor about K_{1} of integral group rings. Apart from avoiding stabilization, reduced norms, etc., the main simplification here consists in sidestepping induced characters ( cf. Lemma 7.1 ). The rest is essentially copied from Bass’s paper [1]. We shall completely abandon our previous notational conventions — they have used up too many letters.

1. Rationality. Let Q be a field . By an algebra over Q we shall always mean a Q-algebra A which is finite dimensional and separable. A set S of subalgebras is called a saturated family if, for each simple component A_{i} of A, the obvious map B → A → A_{i} is surjective for suitable B ∈ S. Example: A = Qπ, where π is a finite abelian group whose order does not vanish in Q, and S = {Qπ′} with π′ running over all cyclic subgroups.

Now let R/Q be an arbitrary field extension. By an R-character of A we shall mean an element of X_{R}(A) = R ⊗_{Z} K_{0}(R ⊗ A), where K_{0}(R ⊗ A) denotes the Grothendieck group of representations of the R-algebra R ⊗ A. An R-character will be said to be Q-rational if it lies in the obvious subspace R ⊗_{Q} X_{Q}(A). The following easy rationality criterion will turn out to be useful.

LEMMA 7.1: An R-character ϕ of A is Q-rational if and only if this is so for all its restrictions ϕ|B to members B of a saturated family of subalgebras of A.

Proof : If A = ∏_{i} A_{i} over Q, then X_{F} (A) = ∏_{i} X_{F} (A_{i}) for both F = Q and F = R. Hence rationality can be tested on the components ϕ_{i} of ϕ corresponding to this decomposition. But each ϕ_{i} also occurs somewhere among the analogous decompositions of the ϕ|B.

2. Units. From now on we take Q and R to be the fields of rationals and reals, respectively.

First let A = K be a field and D be its ring of integers. R ⊗_{Q} K = ∏_{j} L_{j}, where L_{j} ≃ R for j = 1, …, r_{1} and L_{j} ≃ C for j = r_{1} + 1, …, r_{1} + r_{2} in the standard notation. Let V = R^{r_{1}+r_{2}} and let λ : ∏_{j} L_{j}^{×} → V be defined componentwise by λ_{j}(x) = log |x| for j ≤ r_{1} and by λ_{j}(x) = log |x|^{2} for the rest. Then Dirichlet’s Unit Theorem says that λ maps the units of D onto a lattice of corank 1, trivializing only torsion units. We now present Bass’s reformulation of this result.

First we note that, according to Dirichlet, λ yields an R-linear surjection Hom_{R}(V, R) → Hom_{Z}(D^{×}, R) with 1-dimensional kernel. Secondly, mapping the j^{th} irreducible representation of R ⊗_{Q} K to the j^{th} coordinate function of V obviously produces an isomorphism X_{R}(K) → Hom_{R}(V, R).

The composite of these two maps is a surjection X_{R}(K) → Hom_{Z}(D^{×}, R) whose kernel, since it is 1- dimensional, must be R ⊗_{Z} X_{Q}(K). In fact, this composite maps a representation ψ of R ⊗ K to the homomorphism u ↦ log | det ψ(u)|, which is zero for rational ψ, since such ψ is a multiple of the regular representation of of K, and det ψ(u) is a unit in Z.

LEMMA 7.2: Let D be an order in a commutative algebra A over Q. Then

0 → R ⊗ X_{Q}(A) → X_{R}(A) → Hom_{Z}(D^{×}, R) → 0  (∗)

is exact.

Proof: For A = K and D as above, this is just a summary of the preceding arguments. For general A and maximal order D, it results from splitting everything according to the decomposition A = ∏_{i} A_{i} into simple components. Finally, if D is allowed to be an arbitrary order, the term Hom_{Z}(D^{×}, R) does not notice the difference, since D^{×} is of finite index in the units of the maximal order.

THEOREM 7.3: Let D be an order in a commutative Q-algebra A, and let T be a set of subrings of D such that {Q ⊗_{Z} C | C ∈ T } form a saturated family of subalgebras of A. Then {C^{×} | C ∈ T } generates a subgroup of finite index in D^{×}. Proof: We shall prove that Hom_{Z}(D^{×}, R) → ∏_{C} Hom_{Z}(C^{×}, R), as C ranges over T , is injective. Indeed, let h ∈ Hom_{Z}(D^{×}, R) and ϕ ∈ X_{R}(A) be a pre-image of h in the middle term of (*). Then the vanishing of h|C^{×} in Hom_{Z}(C^{×}, R) means that ϕ|R⊗_{Z} C vanishes in X_{R}(Q⊗_{Z} C) and therefore lies in R ⊗_{Z} X_{Q}(Q ⊗_{Z} C), by Lemma 7.2. In other words, ϕ is Q-rational when restricted to every R ⊗_{Z} C, hence is itself Q-rational, by Lemma 7.1; and thus gives to a trivial h. In short, h|C^{×} is trivial for all C only if it was trivial to begin with, Q.E.D.

REMARK: This theorem can be applied to D = Zπ and C = Zπ′, where π is a finite abelian group, and π′ runs over all cyclic subgroups. It implies that the units of the latter generate a subgroup of finite index in the units of the former. However, this quasi-surjectivity of lattices is deduced from the surjectivity of the Euclidean spaces they generate. The crucial intervention of transcendental tools (logarithms), which cannot be arithmetized in an obvious way (say, by p-adic analysis), destroys all arithmetical information about the index.

References.

[1] H. Bass, The Dirichlet Unit Theorem, Induced Characters, and Whitehead Groups of Finite Groups, Topology 4 (1966), 391 – 410.

[2] C. Curtis and I. Reiner, Representation Theory of Finite Groups and Associative Algebras, J. Wiley, New York (1962).

[3] H.Hasse, Zahlentheorie, Akademie Verlag, Berlin (1963).

[4] K. Hoechsmann, S.K. Sehgal, and A. Weiss, Cyclotomic Units and the Unit Group of an Elementary Abelian Group Ring, Arch. d. Math. 45 (1985), 5 – 7.

[5] K. Hoechsmann, S.K. Sehgal, Units in Regular Elementary Abelian Group Rings, Arch. d. Math. 47 (1986), 413 – 417.

[6] K. Hoechsmann, S.K. Sehgal, Units in Regular Abelian p-Group Rings, to appear in J. of Number Theory.

[7] S. K. Sehgal, Topics in Group Rings, Marcel Dekker, New York (1978).

Department of Mathematics, University of British Columbia, Vancouver, B.C., Canada.