Presenting Powers of Augmentation Ideals of Elementary p-Groups

Guoping Tang

Department of Applied Mathematics Northwestern Polytechnical University Xi’an Shaanxi 710072 P.R. China email tanggp@nwpu.edu.cn

Abstract

This note presents powers of the augmentation ideal of an integral group ring of an elementary p-group, generalizing results of Bak and Vavilov for elementary 2-groups and of Parmenter for elementary 3-groups.

2000 AMS Subject Classification 16S34, 20C05 Keywords: integral group ring, augmentation ideal

0 Introduction

Recall that a group is called an elementary p-group where p is a natural prime, if it is abelian of exponent p. Such groups are also called p-elementary.

Let G be a group, ZZG its integral group ring and ΔG the augmentation ideal of ZZG. The n’th power Δ^{n}(G) of ΔG is generated as an abelian group by all products (g_{1} − 1)(g_{2} − 1) ⋯ (g_{n} − 1) where g_{i} ∈ G. In [1], Bak and Vavilov considered the problem of finding a complete set of relations among these generators. The main result in [1] solves this problem when G is an elementary 2-group. In [2], Parmenter solves the problem when either G is an elementary 3-group or G is an elementary p-group where p > 3, but n ≤ 3. The current paper solves the problem when G is an elementary p-group and n is arbitrary.

The rest of the paper is organized as follows. In section 1, we recall well known relations among the generators (g_{1} − 1) ⋯ (g_{n} − 1) of Δ^{n}(G) when G is arbitrary or abelian and provide a family of new relations when G is an elementary p-group, p > 3. In section 2, we show that the relations above account for all relations among the generators of Δ^{n}(G) when G is an elementary p-group.

Acknowledgement * The author gratefully acknowledges the hospitality of Prof. A. Bak and the support of SFB 343, Fakultät für Mathematik, Universität Bielefeld, Germany during the preparation of the present article.

1 Old and New Relations in Δ^{n}(G)

Let G be a group, ZZ[G] its integral group ring and η: ZZ[G] → ZZ, ∑_{g} a_{g}g ↦ ∑_{g} a_{g}, the

augmentation homomorphism. The kernel Δ(G) of the homomorphism η is called the augmentation ideal of ZZ[G]. It is clear that Δ(G) is freely generated as an abelian group by the elements

[g] := g − 1, g ∈ G \ {1}

(cf. [3]). This implies that the n-th power Δ^{n}(G) := Δ(G)^{n} of the augmentation ideal Δ(G) is generated as an abelian group by the products

[g_{1}, …, g_{n}] := (g_{1} − 1) … (g_{n} − 1), g_{1}, …, g_{n} ∈ G.

There are two obvious relations which hold for any group G. The first is the normalizing relation

(N)  [g_{1}, …, g_{n}] = 0 (n ≥ 1), whenever some g_{i} = 1.

The second is that the symbol [g_{1}, g_{2}] is a 2-cocycle:

[g_{2}, g_{3}] − [g_{1}g_{2}, g_{3}] + [g_{1}, g_{2}g_{3}] − [g_{1}, g_{2}] = 0.

This implies the relation

(R)  [g_{1}, …, g_{n}] (n ≥ 2) is a 2-cocycle in g_{i−1}, g_{i}

when the other variables are fixed. If G is abelian then one has in addition to the above the symmetric relation

(S)  [g_{1}, …, g_{i}, …, g_{j}, …, g_{n}] = [g_{1}, …, g_{j}, …, g_{i}, …, g_{n}]  (n ≥ 2)

got by swapping g_{i} with g_{j} and leaving the other elements unchanged. The relations above are of course well known. When G is an elementary p-group, we shall require a new family of relations which we describe next.

For the rest of this section, let p denote some fixed natural prime and m a natural number such that 3 ≤ m ≤ p + 1. We shall define for each m, sequences of integers a_{1}^{(m)}, ⋯, a_{p−m+1}^{(m)} and b_{1}^{(m)}, ⋯, b_{m−2}^{(m)} and use them to describe new relations in Δ^{n}(G) for any n ≥ m and any elementary p-group G. Let X denote an indeterminate and ZZ[X] the integral polynomial ring in X. It is easy to see that for each m, the free ZZ-module of all polynomials f(X) ∈ ZZ[X] such that degree (f(X)) ≤ p − 1 and f(1) = 0 has as ZZ-basis the set

{(X^{i} − 1)(X − 1)^{m−2} | 1 ≤ i ≤ p − m + 1} ∪ {(X − 1)^{i} | 1 ≤ i ≤ m − 2}.

Let F^{(m)}(X) denote the polynomial

F^{(m)}(X) = (X^{p−m+2} − 1)(X − 1)^{m−2} − X^{p} + 1.

We define now the elements a_{i}^{(m)}(1 ≤ i ≤ p − m + 1) and b_{i}^{(m)}(1 ≤ i ≤ m − 2) such that

F^{(m)}(X) = ∑_{i=1}^{p−m+1} a_{i}^{(m)}(X^{i} − 1)(X − 1)^{m−2} + ∑_{i=1}^{m−2} b_{i}^{(m)}(X − 1)^{i}.

Let ∂^{i}F^{(m)}(X) denote the i’th derivative of F^{(m)}(X). For 1 ≤ i ≤ m − 2, one computes that ∂^{i}F^{(m)}(1) = −p(p − 1) ⋯ (p − i + 1) = b_{i}^{(m)}(i!). Thus b_{i}^{(m)} = −\binom{p}{i}. We use the computation of the b_{i}^{(m)}’s to define a further sequence b_{i,1}^{(m)}, ⋯, b_{i,p−1}^{(m)} of integers. Let ⟨h⟩ denote a group of order p generated by h. Since 1 ≤ i ≤ m − 2 < p, p divides \binom{p}{i}. Thus b_{i}^{(m)}(h − 1) ∈ Δ^{p}(⟨h⟩) by [3, Lemma 6.10]. In particular b_{i}^{(m)}(h − 1) ∈ Δ^{m−i}(⟨h⟩). One checks straightforward that the free ZZ-module Δ^{m−i}(⟨h⟩) has the ZZ-basis

{(h − 1)^{m−i−1}(h^{j} − 1) | 1 ≤ j ≤ p − 1}.

Define the integers b_{i,j}^{(m)} (1 ≤ i ≤ m − 2, 1 ≤ j ≤ p − 1) by

−\binom{p}{i}(h − 1) = ∑_{j=1}^{p−1} b_{i,j}^{(m)}(h − 1)^{m−i−1}(h^{j} − 1).

Note that if we let h = 1, the equation above holds trivially.

Let G be an elementary p-group. It follows from the definitions above that for any element g ∈ G, we have the identity

(g^{p−m+2} − 1)(g − 1)^{m−2} = ∑_{i=1}^{p−m+1} a_{i}^{(m)}(g^{i} − 1)(g − 1)^{m−2} + ∑_{i=1}^{m−2} b_{i}^{(m)}(g − 1)^{i}

in Δ^{m−1}(G). If h ∈ G then multiplying both sides of the identity above by (h − 1), we get the equation

(g^{p−m+2} − 1)(g − 1)^{m−2}(h − 1) = ∑_{i=1}^{p−m+1} a_{i}^{(m)}(g^{i} − 1)(g − 1)^{m−2}(h − 1) +

∑_{i=1}^{m−2} ∑_{j=1}^{p−1} b_{i,j}^{(m)}(g − 1)^{i}(h − 1)^{m−i−1}(h^{j} − 1)

in Δ^{m}(G), i.e.

[g^{p−m+2}, \underbrace{g, ⋯, g}_{m−2}, h] = ∑_{i=1}^{p−m+1} a_{i}^{(m)}[g^{i}, \underbrace{g, ⋯, g}_{m−2}, h] +

∑_{i=1}^{m−2} ∑_{j=1}^{p−1} b_{i,j}^{(m)}[\underbrace{g, ⋯, g}_{i}, \underbrace{h, ⋯, h}_{m−i−1}, h^{j}].

Therefore for any n ≥ m, we get the relation

[g^{p−m+2}, \underbrace{g, ⋯, g}_{m−2}, h, g_{m+1}, ⋯, g_{n}] = (T_{m})

∑_{i=1}^{p−m+1} a_{i}^{(m)}[g^{i}, \underbrace{g, ⋯, g}_{m−2}, h, g_{m+1}, ⋯, g_{n}] +

∑_{i=1}^{m−2} ∑_{j=1}^{p−1} b_{i,j}^{(m)}[\underbrace{g, ⋯, g}_{i}, \underbrace{h, ⋯, h}_{m−i−1}, h^{j}, g_{m+1}, ⋯, g_{n}]

in Δ^{n}(G).

Setting m = 3 and p = 2 in the relation (T_{m}), we get the relation (T) in [1], in the equivalent form found in Lemma 3.1 of [1]. Setting m = 3 and letting p ≥ 3 in the relation (T_{m}), we get the relation denoted by (T_{p}) in [2].

Suppose p = 3 and n ≥ 4. Then there is the relation

[g^{2}, g, g, h, −] = [g, h, h, h^{2}, −]

in Δ^{n}(G), which is denoted by (U) in [2]. This relation can be obtained from our relations (T_{3}) and (T_{4}) as follows. By our (T_{3}),

[g^{2}, g, g, h, −] = −[g, g, g, h, −] + [g, g, h, h^{2}, −] + [g, g, h, h, −]

and by our (T_{4}),

[g, g, g, h, −] = −[g, h, h, h, −] + [g, g, h, h^{2}, −] + [g, g, h, h, −].

So we get relation (U). Conversely since our (T_{3}) coincides with relation (T_{3}) in [2] when p = 3, it follows that our (T_{4}) can be obtained from relations (T_{3}) and (U) in [2].

2 A Presentation of Δ^{n}(G) for G p-Elementary

The goal of this section is to prove the following result.

Theorem 2.1 Let G be an elementary p-group. Then the n’th power Δ^{n}(G) of the augmentation ideal ΔG of the integral group ring ZZG is a free abelian group on the symbols [g_{1}, ⋯, g_{n}] := (g_{1} − 1) ⋯ (g_{n} − 1), g_{i} ∈ G, modulo the relations

(N)  if n = 1
(N), (R), (S)  if n = 2
(N), (R), (S), (T_{3}), ⋯, (T_{n})  if 3 ≤ n ≤ p
(N), (R), (S), (T_{3}), ⋯, (T_{p}), (T_{p+1}) if p < n.

Proof It suffices to prove the theorem in the special case that the order |G| of G is finite. The proof will be by induction on |G|.

Suppose |G| = p. Let g be a generator of G. Thus G = ⟨g⟩. One verifies easily that B^{n}(⟨g⟩) := \{[\underbrace{g, ⋯, g}_{n−1}, g^{j}] | 1 ≤ j ≤ p − 1\} is a ZZ-basis for Δ^{n}(G). It suffices to show

that any generator [g^{i_{1}}, ⋯, g^{i_{n}}] of Δ^{n}(G) can be exhibited using just the relations in the theorem, as an integral linear combination of the basis elements above. If n = 1, the result is trivial. We proceed now by induction on n. Suppose n > 1. Suppose that one of the exponents i_{j} is 1. Applying (S), we can assume that j = 1, i.e. i_{1} = 1. By the induction assumption, we can write [g^{i_{2}}, ⋯, g^{i_{n}}] as a linear combination of the elements \{[\underbrace{g, ⋯, g}_{n−2}, g^{j}] | 1 ≤ j ≤ p − 1\}. Thus [g, g^{i_{2}}, ⋯, g^{i_{n}}] is a linear combination of the

elements in \{[\underbrace{g, ⋯, g}_{n−1}, g^{j}] | 1 ≤ j ≤ p − 1\}. We proceed now by induction on i_{1}. Suppose

i_{1} > 1. By (R), we can write [g^{i_{1}}, ⋯, g^{i_{n}}] = [g^{i_{1}−1}, g^{i_{2}}, ⋯, g^{i_{n}}]+[g, g^{i_{1}+i_{2}−1}, g^{i_{3}}, ⋯, g^{i_{n}}]− [g, g^{i_{1}−1}, g^{i_{3}}, ⋯, g^{i_{n}}]. By the induction assumption, we can write each of the generators on the right hand side of the equation above, as a linear combination of the basis elements. This completes the proof when |G| = p.

Suppose |G| > p. Write G = ⟨g⟩ ⊕ H where g is an element of G of order p. Let B^{n}(⟨g⟩) denote the ZZ-basis above for Δ^{n}(⟨g⟩). For 1 ≤ j ≤ n, let B^{j}(H) and B^{j}(G) denote ZZ-bases of Δ^{j}(H) and Δ^{j}(G), respectively. It follows easily from Theorem 2 in [2] that we can choose the B^{j}(H)’s and B^{j}(G)’s such that the following holds:

(1) If n = 1,  B(G) = {[g^{i}h] | 0 ≤ i ≤ p − 1, h ∈ H}\{[1]}.

(2) If 2 ≤ n ≤ p, B^{n}(G) = ∪_{i=0}^{n−2}[\underbrace{g, ⋯, g}_{i}]B^{n−i}(H) ∪ ∪_{i=1}^{p−n+1}[g^{i}, \underbrace{g, ⋯, g}_{n−2}]B^{1}(H) ∪ B^{n}(⟨g⟩).

(3) If p < n,  B^{n}(G) = ∪_{i=0}^{p−1}[\underbrace{g, ⋯, g}_{i}]B^{n−i}(H) ∪ B^{n}(⟨g⟩).

If n = 1, the conclusion of the theorem is trivial. No induction on |G| is required. So we suppose for the rest of the proof that n ≥ 2.

A generator [g_{1}, ⋯, g_{n}] will be called good if one of the following holds:

(i) Each g_{i} (1 ≤ i ≤ n) is a power of g.

(ii) 2 ≤ n ≤ p and each g_{i} (1 ≤ i ≤ n) has the property that either g_{i} = g or g_{i} ∈ H.

(iii) 2 ≤ n ≤ p and for some permutation π of \{1, ⋯, n\}, [g_{π(1)}, ⋯, g_{π(n)}] = [g^{i}, \underbrace{g, ⋯, g}_{n−2}, h]

for some 1 ≤ i ≤ p − n + 1 and h ∈ H.

(iv) p < n and for some permutation π of \{1, ⋯, n\}, [g_{π(1)}, ⋯, g_{π(n)}] = [\underbrace{g, ⋯, g}_{k}, h_{k+1}, ⋯, h_{n}]

where 0 ≤ k ≤ p − 1 and h_{k+1}, ⋯, h_{n} ∈ H.

Since our basis B^{n}(G) satisfies (i)-(iv) above and since |H| < |G|, it follows from the induction assumption on H and the case G = ⟨g⟩ above that any good generator can be exhibited using only the relations in the theorem, as a linear combination of the elements of B^{n}(G). Thus it suffices now to show that an arbitrary generator can be exhibited using only the relations of the theorem as a linear combination of good generators.

We claim that the relation (R) along with the relation (S) allows one to write any generator as a linear combination of generators [g_{1}, ⋯, g_{n}] such that each g_{j} is either a power of g or is in H. Let [g^{i}h_{1}, g^{j}h_{2}, −] denote a typical generator. The relation (R) shows that

[g^{i}h_{1}, g^{j}h_{2}, −] = [g^{i+j}h_{1}, h_{2}, −] + [g^{i}h_{1}, g^{j}, −] − [g^{j}, h_{2}, −].

Applying (R) to the first 2 summands on the right hand side, we obtain

[g^{i+j}h_{1}, h_{2}, −] = [g^{i+j}, h_{1}h_{2}, −] + [h_{1}, h_{2}, −] − [g^{i+j}, h_{1}, −]

and

[g^{i}h_{1}, g^{j}, −] = [h_{1}, g^{i+j}, −] + [g^{i}, g^{j}, −] − [h_{1}, g^{i}, −].

The claim follows from the 3 equations above and (S). Applying (R) to a generator [g^{i}, g^{j}, −], we obtain

[g^{i}, g^{j}, −] = [g^{i+j−1}, g, −] + [g^{i}, g^{j−1}, −] − [g^{j−1}, g, −].

It follows now that any generator of Δ^{n}(G) (n ≥ 2) can be written using only the relations (R) and (S) as a linear combination of generators of the form [\underbrace{g^{i}, g, ⋯, g}_{k}, h_{k+1}, ⋯, h_{n}]

where 0 ≤ i ≤ p − 1 and 0 ≤ k ≤ n. Moreover from (N), it follows that we can assume additionally that 1 ≤ i ≤ p − 1. Generators of this kind will be called nice generators and the pair (i, k) is called the type of the generator.

Suppose n = 2. Then it is easy to check that each nice generator is a good one also. So the proof is complete in this case.

Suppose n ≥ 3. Let [\underbrace{g^{i}, g, ⋯, g}_{k}, h_{1}, ⋯, h_{n−k}] be a nice generator such that n−k ≥ 2.

To shorten notation, let (g^{i}, h_{1}, h_{2}) = [\underbrace{g^{i}, g, ⋯, g}_{k}, h_{1}, ⋯, h_{n−k}]. Applying (R) and (S)

several times, we can write (g^{i}, h_{1}, h_{2}) as linear combination

(g^{i}, h_{1}, h_{2}) = (h_{1}, g^{i−1}g, h_{2}) = (h_{1}g^{i−1}, g, h_{2}) +
(h_{1}, g^{i−1}, h_{2}) − (g^{i−1}, g, h_{2}) = (g^{i−1}, g, h_{1}h_{2}) +

(g, h_{1}, h_{2}) − (g^{i−1}, g, h_{1}) + (g^{i−1}, h_{1}, h_{2}) − (g^{i−1}, g, h_{2}) = (∗)

(g^{i−1}, g, h_{1}h_{2}) + (g^{i−1}, h_{1}, h_{2}) + (g, h_{1}, h_{2}) −
(g^{i−1}, g, h_{1}) − (g^{i−1}, g, h_{2}).

Suppose n = 3. Here the only nice generators which are not good generators are those of the kind [g^{i}, h_{1}, h_{2}] (2 ≤ i ≤ p−1) and [g^{p−1}, g, h]. By (∗), [g^{i}, h_{1}, h_{2}] is a sum of good generators and by T_{3}, [g^{p−1}, g, h] is a sum of good generators. So the proof is complete in this case.

Suppose p < n. The only nice generators x = [\underbrace{g^{k}, g, ⋯, g}_{l}, h_{l+1}, ⋯, h_{n}] which are not

good are those whose type (k, l) satisfies one of the following:

(a) 1 < k ≤ p − 1 and n − l ≥ 1.

(b) k = 1 and p ≤ l ≤ n − 1.

Suppose x satisfies (a) and n − l > 1. By (∗), we can write x as a sum of good generators, nice generators of type (k − 1, l′), and nice generators satisfying (b). Thus by several applications of (∗), we can reduce case (a) to the case such that 1 < k ≤ p − 1 and n − l = 1. In this case, l − 1 = n − 2 ≥ p − k so by relation (T_{p−k+2}), we can write x as a sum of good generators, nice generators of type (k′, l′) with k′ < k, and nice generators satisfying (b). Thus by several applications of (∗) as well as (T_{p−k+2}), …, (T_{3}), we can reduce case (a) to case (b). Suppose x satisfies case (b). By relation (T_{p+1}), we can write x as a sum of good generators and nice generators of type (1, i), i < l. Thus by several applications of (T_{p+1}), we can write x as a sum of good generators and nice generators of type (1, i), i ≤ p − 1. Since such generators are good, the proof is complete in the case p < n.

It is easy to see that we have already give a complete proof for cases p = 2 and p = 3. So we now assume that p > 3. It remains to treat the case 4 ≤ n ≤ p.

Suppose p > 3 and 4 ≤ n ≤ p. Here the only nice generators x = [\underbrace{g^{k}, g, ⋯, g}_{l}, h_{l+1}, ⋯, h_{n}]

which are not good are those whose type (k, l) satisfies one of the following:

(c) 1 < k ≤ p − 1 and 1 ≤ l < n − 1.

(d) p − n + 1 < k ≤ p − 1 and l = n − 1.

If (k, l) satisfies (c) then by applying (∗), we can write x as a sum of nice generators of types (k − 1, l), (k − 1, l + 1), and (1, l). Applying this procedure over and over again to the nice generators obtained each time, we obtain that x is a sum of nice generators each of which is either good or satisfies (d). Thus it suffices to show that if x is a nice generator whose type satisfies (d) then it is a sum of good generators. Let x be a nice generator whose type (k, l) satisfies (d). Choose m such that k = p − m + 2. Since k > p − n + 1, it follows that m ≤ n. Applying relation (T_{m}) to x, we can write x as a sum of good generators and nice generators of type (i, n − 1), i < k. Iterating this procedure, we can write using relations (T_{m}), ⋯, (T_{n}), x as a sum of good generators and nice generators of type (p − n + 1, n − 1). Since nice generators of type (p − n + 1, n − 1) are good, the proof is complete.

References

[1] Anthony Bak and Nikolai Vavilov, Presenting powers of augmentation ideals and Pfister forms, K-Theory 20 (2000), 299 – 309 [2] M. M. Parmenter, A basis for powers of the augmentation ideal, Algebra Colloquium,

to appear

[3] I. B. S. Passi, Group rings and their augmentation ideals, Lecture Notes in Math. Vol 715 (1979)