The second Fox quotient of arbitrary group rings

by Manfred Hartl

Institut de Recherche Mathématique Avancée, ULP et CNRS, 7 rue René-Descartes, 67084 Strasbourg Cedex, FRANCE

Let G be a group, ZZ(G) be the integral group ring of G and I(G) the augmentation ideal of ZZ(G). Let H be a subgroup of G and J be a left-ideal of ZZ(H). Then it is a classical problem to study the k-th generalized Fox quotient I^{k−1}(G)J/I^{k}(G)J (usually one considers J = I(H)), for k ≥ 1 (setting I^{o}(G) = ZZ(G)). The first Fox quotient was determined a long time ago [12], and the related quotient I(G)I(H)^{k−1}/I(G)I(H)^{k} was recently determined by Karan– Vermani [7]. But only partial results are known on the second Fox quotient (and some higher ones), where notably H is always supposed to be a semi-direct factor of G, see [4], [5], [6], [8], [9]. As a striking fact, in all known cases a reduction of the concerned quotient was obtained to sums and tensor products of certain subquotients of the group rings of G and H separately (or of the semidirect complement of H in G). Introducing a homological approach to the problem, we here give such a reduction of I(G)J/I^{k}(G)J for arbitrary G, H, J and k > 1 as above. Using this result the second generalized Fox quotient is now completely determined in terms of G and some filtration quotients of J, or directly in terms of G and H in the classical case J = I(H). By the same argument, we also obtain a unifying new proof of the known results on the first Fox quotient and on I(G)J/I(G)I(H)J. The main theorem 3 will also be further exploited in subsequent studies of higher Fox quotients and Fox subgroups in [2], [3].

Convention. In this paper, maps denoted by i or j (possibly indexed) are always induced by the inclusion i: I(H) ↪ I(G), and q (possibly indexed) always denotes a canonical quotient map. Moreover, maps denoted by g, g′ or h, h′ are always induced by multiplication in ZZ(G) or ZZ(H), respectively. Moreover, we write G_{i} for the i-th term of the lower central series, with G_{1} = G.

We start by quoting slight generalizations of some known results which will be given a new and easy proof here.

Research supported by an individual fellowship of the Human Capital and Mobility Programme of the European Union.

Theorem 1 The first generalized Fox quotient is reduced to J by the natural isomorphism

\frac{ZZ(G)J}{I(G)J} ≅ \frac{J}{I(H)J} . □

We note that the identity G∩(1+I(G)J) = H ∩(1+I(H)J) in [7] is immediate from this isomorphism combined with the classical identity G∩(1+J) = H∩(1+J), see [11] (which also can be easily derived from lemma 9 below).

Theorem 2 (cf. [7]) There is a (non-canonical) isomorphism

\frac{I(G)J}{I(G)I(H)J} ≅ \frac{I(H)J}{I^{2}(H)J} ⊕ (\frac{I(G)}{ZZ(G)I(H)}) ⊗ (\frac{J}{I(H)J} ) . □

Now we turn to our key result. Let k > 1 and consider the following commutative square.

\begin{matrix}(\frac{I(H)}{I^{k}(H)}) ⊗_{H} (\frac{J}{I^{k−1}(H)J} ) & ⟶^{i⊗id} & (\frac{I(G)}{I^{k}(G)}) ⊗_{H} (\frac{J}{I^{k−1}(H)J} ) \\ ↓^{↓^{h}} & & ↓^{↓^{g}} \\ \frac{I(H)J}{I^{k}(H)J} & ⟶^{j} & \frac{I(G)J}{I^{k}(G)J} \end{matrix}  (1)

Theorem 3 Diagram (1) is a cocartesian square of abelian groups; in other words, the following sequence is exact:

Ker(h) ⟶^{i⊗id|Ker(h)} (\frac{I(G)}{I^{k}(G)}) ⊗_{H} (\frac{J}{I^{k−1}(H)J} ) ⟶^{g} \frac{I(G)J}{I^{k}(G)J} → 0  (2)

This also implies the relation

Ker(j) = h Ker(i ⊗ id) . □

In order to derive an exact sequence for I(G)J/I^{k}(G)J consider the connecting homomorphism

E_{∗}: T =^{def} Tor_{1}^{H}(\frac{I(G)}{I(H) + I^{k}(G)} , \frac{J}{I^{k−1}(H)J} ) → (\frac{I(H) + I^{k}(G)}{I^{k}(G)}) ⊗_{H} (\frac{J}{I^{k−1}(H)J} )

which arises from applying the functor − ⊗_{H} (J/I^{k−1}(H)J) to the short exact sequence

0 → \frac{I(H) + I^{k}(G)}{I^{k}(G)} ⟶^{j} \frac{I(G)}{I^{k}(G)} ⟶^{q} \frac{I(G)}{I(H) + I^{k}(G)} → 0 .

Corollary 4 There is a natural exact sequence

T ⟶^{h′E_{∗}} \frac{I(H)J + I^{k}(G)J}{I^{k}(G)J} ⟶^{j} \frac{I(G)J}{I^{k}(G)J} ⟶^{q̃} (\frac{I(G)}{I(H) + I^{k}(G)}) ⊗_{H} (\frac{J}{I^{k−1}(H)J} ) → 0

where q̃ is well defined by requiring q̃g = q ⊗ id. □

For k = 2 we use the natural identification ϕ: G/G_{2} ≅ I(G)/I^{2}(G), aG_{2} ↦ a − 1 + I^{2}(G) for a ∈ G, to derive from theorem 3 the following description of the second generalized Fox quotient.

Consider the following diagram, where the tensor products are now taken over the ring of integers.

T ′ ⟶^{τ} (\frac{HG_{2}}{G_{2}} ) ⊗ (\frac{J}{I(H)J} ) ⟶^{ι⊗id} (\frac{G}{G_{2}} ) ⊗ (\frac{J}{I(H)J} ) ⟶^{π⊗id} (\frac{G}{HG_{2}} ) ⊗ (\frac{J}{I(H)J} )
∥  \begin{matrix}↓ & h(ϕ⊗id) \\ ↓ \end{matrix}  \begin{matrix}↓ & g(ϕ⊗id) \\ ↓ \end{matrix}  ∥

T ′  ⟶^{hτ} \frac{I(H)J}{I(H ∩ G_{2})J + I^{2}(H)J} ⟶^{j}  \frac{I(G)J}{I^{2}(G)J}  ⟶^{q̃} (\frac{G}{HG_{2}} ) ⊗ (\frac{J}{I(H)J} )

Here T′ = Tor_{1}^{Z Z}(G/HG_{2}, J/I(H)J), and the top row is part of the corresponding 6-term exact sequence where ι is induced by the inclusion H ↪ G and π is the canonical quotient map. Moreover, q̃ is well defined as a factorization of π ⊗ id by theorem 3.

Then I(G)J/I^{2}(G)J is functorially determined by

Corollary 5 The rows of the diagram above are exact; and omitting the terms T′, the bottom sequence is induced by the top sequence via the map h(ϕ ⊗ id).

More explicitely, Ker(j) is generated by the cosets {(h − 1)x} where h ∈ H, x ∈ J such that there is k ∈ ZZ and an element g ∈ G for which kx ∈ I(H)J and g^{k} lies in the coset hG_{2}. □

The latter description of Ker(j) is obtained by using the canonical generators of the torsion product, see [10] V.6.

Corollary 6 If the sequence

1 → HG_{2}/G_{2} → G/G_{2} → G/HG_{2} → 1

of abelian groups splits then

\frac{I(G)J}{I^{2}(G)J} ≅ \frac{I(H)J}{I(H ∩ G_{2})J + I^{2}(H)J} ⊕ (\frac{G}{HG_{2}} ) ⊗ (\frac{J}{I(H)J} ) . □

This last result in particular applies if H is a semidirect factor of G (i.e. if either G = H K or G = K H for some subgroup K). Thus we rediscover the results of Khambadkone and Karan-Vermani in [5], [6], [8], [9].

In the case J = I(H) we can improve corollary 5, as follows. Recall the exterior square A^{∧ 2} of abelian groups A defined by A^{∧ 2} = (A ⊗ A)/Δ where Δ is the subgroup generated by the elements a ⊗ a, a ∈ A. Consider the commutative square

\begin{matrix}(H/H_{2})^{∧ 2} & ⟶^{l_{G,H}} & (G/G_{2}) ⊗ (H/H_{2}) \\ ↓^{↓^{c}} & & ↓^{↓^{μ}} \\ H_{2}/H_{3} & ⟶^{d} & I(G)I(H)/I^{2}(G)I(H) \end{matrix}  (3)

where for a, b, h ∈ H, g ∈ G and x ∈ H_{2}

l_{G,H}((aH_{2})∧(bH_{2})) = (aG_{2})⊗(bH_{2})−(bG_{2})⊗(aH_{2}), c((aH_{2})∧(bH_{2})) = aba^{−1}b^{−1}H_{3},

μ((gG_{2}) ⊗ (hH_{2})) = (g − 1)(h − 1) + I^{2}(G)I(H) , d(xH_{3}) = x − 1 + I^{2}(G)I(H) .

Corollary 7 The following sequence is exact:

Ker(c) ⟶^{l_{G,H}|Ker(c)} (G/G_{2}) ⊗ (H/H_{2}) ⟶^{μ} I(G)I(H)/I^{2}(G)I(H) → 0 □

This follows from theorem 3 by setting J = I(H) and k = 2, since then Ker(h) = (ϕ⊗ϕ)l_{H,H}Ker(c) by the isomorphism I^{2}(H)/I^{3}(H) ≅ U_{2}L(G) obtained in [1].

Remark 8 Modulo H_{3}, the group Ker(d) equals the second Fox subgroup G ∩ (1 + I^{2}(G)I(H)). But by corollary 7 we have Ker(d) = cKer(l_{G,H}) in (3), which provides the starting point of our homological calculation of the second Fox subgroup in [3].

Now we turn to the proofs. The step into homological algebra is provided by the following elementary

Lemma 9 The canonical map

μ_{J} : ZZ(G) ⊗_{H} J → → ZZ(G)J

is an isomorphism.

Proof: Using the inclusion η: J ↪ ZZ(H) we have the following commutative square

ZZ(G) ⊗_{H} J  ⟶^{μ_{J}} ZZ(G)J
↓^{id⊗η}  ↓
ZZ(G) ⊗_{H} ZZ(H) ≅ ZZ(G)

But id⊗η is injective since ZZ(G) is is a free H -module, so also μ_{J} is injective. □

In order to describe our key lemma, let U ⊂ M ⊂ I(G) be right and V ⊂ N ⊂ J be left H -submodules such that U ⊂ I(H). Consider the following commutative square.

\begin{matrix}\frac{I(H) ⊗_{H} J}{U ⊗_{H} V} & ⟶^{i⊗id} & \frac{I(G) ⊗_{H} J}{M ⊗_{H} N} \\ ↓^{↓^{h}} & & ↓^{↓^{g}} \\ \frac{I(H)J}{UV} & ⟶^{j_{o}} & \frac{I(G)J}{MN} \end{matrix}  (4)

Lemma 10 In (4), Ker(g) = (i ⊗ id)Ker(h) .

Proof of lemma 10 and theorem 1: Consider the following diagram.

I(G)J  ↪  ZZ(G)J  ⟶ \frac{ZZ(G)J}{I(G)J}
↑^{g′} ↑  ≅ ↑^{μ_{J}}  ↑μ̄_{J}
Tor_{1}^{H}(ZZ, J) ⟶^{E_{∗}} I(G) ⊗_{H} J ⟶^{ν⊗id} ZZ(G) ⊗_{H} J ⟶^{ϵ⊗id} ZZ ⊗_{H} J
∥  ↑^{i⊗id}  ↑^{i⊗id}
Tor_{1}^{H}(ZZ, J) ⟶^{E_{∗}} I(H) ⊗_{H} J ⟶^{ν⊗id} ZZ(H) ⊗_{H} J ≅^{h′}  J

The two upper rows are exact in the third column, with I(G) ↪^{ν} ZZ(G) ⟶^{ϵ} ZZ being the canonical inclusion and the augmentation map. Since μ_{J} is isomorphic by lemma 9 and since g′ is surjective, the induced map μ̄_{J} is an isomorphism, too. But ZZ ⊗_{H} J ≅ J/I(H)J, which proves theorem 1.

The lower two rows are exact in the second column when we take the maps E_{∗} to be connecting homomorphisms in the appropriate long exact sequences; by their naturality the diagram commutes. Whence

Ker(μ_{J}(ν ⊗ id)) = Im(E_{∗}) = (i ⊗ id)Im(E_{∗}) = (i ⊗ id)Ker(h′(ν ⊗ id)) .

So in (4),

Ker(g) ≡ Ker(g′) mod (M ⊗_{H} N)
= Ker(μ_{J}(ν ⊗ id))
= (i ⊗ id)Ker(h′(ν ⊗ id))
≡ (i ⊗ id)Ker(h) ,

as was to be shown. □

Proof of theorem 3: Take U = I^{k}(H), M = I^{k}(G), V = N = J in (4). Then

\frac{I(G) ⊗_{H} J}{I^{k}(G) ⊗_{H} J} ≅ (\frac{I(G)}{I^{k}(G)}) ⊗_{H} J ≅ (\frac{I(G)}{I^{k}(G)}) ⊗_{H} (\frac{J}{I^{k−1}(H)J} ) ,

similarly for G being replaced by H. Using this it is easy to identify (1) with (4), so lemma 10 implies the exactness of sequence (2). The remaining statements are immediate consequences of this fact, cf. the diagram above 5. □

Proof of theorem 2: Take U = I(H), M = I(G), V = N = I(H)J in (4). Then by similar arguments as in the proof of theorem 3, (4) is identified with the left-hand square of the following commutative diagram.

\frac{I(H)}{I^{2}(H)} ⊗ \frac{J}{I(H)J} ⟶^{i_{1}⊗id} \frac{I(G)}{I(G)I(H)} ⊗ \frac{J}{I(H)J} ⟶^{q_{1}⊗id} \frac{I(G)}{ZZ(G)I(H)} ⊗ \frac{J}{I(H)J}
↓^{↓^{h}}  ↓^{↓^{g}}  ∥

\frac{I(H)J}{I^{2}(H)J}  ⟶^{j_{1}}  \frac{I(G)J}{I(G)I(H)J}  ⟶^{q̃_{1}}  \frac{I(G)}{ZZ(G)I(H)} ⊗ \frac{J}{I(H)J}

Hence by lemma 10 a homomorphism q̃_{1} exists such that the right-hand square commutes, and the bottom sequence is induced by the top one. But the sequence

0 → \frac{I(H)}{I^{2}(H)} ⟶^{i_{1}} \frac{I(G)}{I(G)I(H)} ⟶^{q_{1}} \frac{I(G)}{ZZ(G)I(H)} → 0

of abelian groups is exact by the classical Fox theorem (i.e. theorem 1 with J = I(H)), and is split; for I(G)/(ZZ(G)I(H)) is a subgroup of ZZ(G)/(ZZ(G)I(H)) ≅ ZZ(G) ⊗_{H} ZZ which is a free abelian group since ZZ(G) is a free H -module. Thus the top sequence in the above diagram is split short exact, whence so is the bottom sequence, as was to be shown. □

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