The non-abelian tensor product of normal crossed submodules of groups

In this article, the notions of non-abelian tensor and exterior products of two normal crossed submodules of a given crossed module of groups are introduced and some of their basic properties are established. In particular, we investigate some common properties between normal crossed modules and their tensor products, and present some bounds on the nilpotency class and solvability length of the tensor product, provided such information is given at least on one of the normal crossed submodules.


Introduction
The notion of the non-abelian tensor product of groups was introduced by Brown and Loday [5,6] following ideas of Miller [14], Dennis [8], and has arisen from applications in homotopy theory of a generalized Van Kampen theorem. Group theorical aspects of this concept have been studied extensively by several authors (see [2,3,4,9,16,17,22]). Especially, one of the main themes of the group theoretical part of research on non-abelian ten-∂ γ 1 = γ 2 ∂ and γ 1 ( g t) = γ 2 (g) γ 1 (t) for all g ∈ G, t ∈ T .
Taking objects and morphisms as defined above, we obtain the category of crossed modules. In this category one can find the familiar notions of injection, surjection, (normal) subobject, kernel, cokernel, exact sequence, etc.; most of them can be found in detail in [11,18].
Let (T, G, ∂) be a crossed module with normal crossed submodules (S, H, ∂) and (L, K, ∂). The following is a list of notations which will be used: where G ab = G/G and ∂ is induced by ∂. We say a crossed module (T, G, ∂) is finite if the groups T and G are both finite. Also, the crossed module (T, G, ∂) is nilpotent (respectively, solvable) when there is n ≥ 1 such that γ n+1 (T, G, ∂) = 1 (respectively, (T, G, ∂) (n) = 1). The smallest n with this property is called the nilpotency class (respectively, solvability length) of (T, G, ∂). Especially, the nilpotent crossed module of class 1 is abelian and the solvable crossed module of length 2 is metabelian.
Let (S, G, ∂) and (L, G, σ) be two crossed modules. There are actions of S on L and of L on S given by s l = ∂(s) l and l s = σ(l) s. We take S (and L) to act on itself by conjugation. The non-abelian tensor product S ⊗ L is defined in [6] as the group generated by symbols s ⊗ l for s ∈ S, l ∈ L, subject to the relations ss ⊗ l = ( s s ⊗ s l)(s ⊗ l), (2.1) 2) for all s, s ∈ S and l, l ∈ L. Note that the identity homomorphism id G : G −→ G is a crossed module with G acting on itself by conjugation, so we can always form the tensor products S ⊗ G, L ⊗ G and G ⊗ G.
Let S and L be as above, and let T be a group. A function φ : S × L −→ T is called a crossed pairing if φ(ss , l) = φ( s s , s l)φ(s, l) and φ(s, ll ) = φ(s, l)φ( l s, l l ) for all s, s ∈ S, l, l ∈ L. It is apparent that the function S × L −→ S ⊗ L, (s, l) −→ s ⊗ l, is the universal crossed paring in the sense that any crossed pairing φ : S×L −→ T determines a unique homomorphism Let S L denote the subgroup of S ⊗ L generated by the elements s ⊗ l with ∂(s) = σ(l). This is a normal subgroup of S ⊗ L and, following [6], the non-abelian exterior product S ∧ L is defined to be the quotient S ⊗ L/S L.
The following proposition summarizes the rather elementary properties of the non-abelian tensor product, the proof of which is left to the reader (see also [4,6]). Proposition 2.1. With the above assumptions and notations, we have (i) If s l = l, l s = s for all s ∈ S, l ∈ L, then S ⊗ L ∼ = S ab ⊗ L ab , where the right-hand side of the isomorphism denotes the usual tensor product of abelian groups.
(ii) There is an isomorphism (iv) These homomorphisms are crossed modules in which the action of G on S ⊗ L is given by g (s ⊗ l) = g s ⊗ g l, and S and L act on S ⊗ L via ∂ and σ. ( for all x ∈ S ⊗ L, s ∈ S, l ∈ L and thus the actions of S on ker λ L , L on ker λ S are trivial. (vi) S L ⊆ ker λ S ∩ ker λ L , whence S L ⊆ Z(S ⊗ L), and S, L act trivially on S L. In particular, for any s ⊗ l ∈ S L, s ⊗ l = s −1 ⊗ l −1 .
Here Γ(S ab ) denotes J.H.C. Whitehead's universal quadratic functor [23], defined for each abelian group A = S ab as the abelian group generated by symbols γ(a) for a ∈ A, subject to the relations γ(a −1 ) = γ(a), for all a, b, c ∈ A. Note that the last condition yields that the map ∆γ : Lemma 2.2. With the above assumptions and notations, we have (i) For all s, s ∈ S, l, l ∈ L, x, y ∈ S ⊗ L, the following identities hold in S ⊗ L: (2.7) (ii) For any s ⊗ l, s ⊗ l ∈ S L, (s ⊗ l)(s ⊗ l ) ∈ S L and ss ⊗ ll = (s ⊗ l)(s ⊗ l )(s ⊗ l)(s ⊗ l ).
Proof. Parts (i) and (ii) are found in [4; Proposition 3] and in the proof of [6; Theorem 2.12], respectively. (iii) Taking into account that any element of [G, S] is written as a finite product of the elements of the form ( g ss −1 ) +1 , where g ∈ G, s ∈ S, and using Proposition 2.1(vi), we need only verify the result for the case y = g ss −1 . Applying (2.4), we have which gives the result.
The following lemmas play a crucial role in our investigation.
Lemma 3.1. With the above assumptions and notations, we have (i) for any x ∈ S ⊗K, y ∈ H ⊗L and z ∈ (S ⊗K)∪(H ⊗L), ∂⊗id K (x) z = λ S (x) z and id H ⊗∂(y) z = λ H (y) z.
(ii) For any x ∈ S ⊗ L and y ∈ S ⊗ K, ∂⊗id L (x) y = id S ⊗∂(x) y.
Proof. Parts (i)-(iv) immediately follow from the definitions of the actions and the properties of crossed module ∂.
(v) Applying the relations (2.4), (2.5) and using the action of T on G, we have For the proof of the second formula, assuming l 0 = h ll −1 and using again the relations (2.3)-(2.5) we see that The proof is complete.
Lemma 3.2. With the above assumptions and notations, we have In particular, α( We now prove that for all elements x ∈ S ⊗ L and (y, Hence, thanks to Lemma 3.1(iii).
In the above lemma, it is easy to see that Imα lies in the kernel of β. So, if we denote by Cokerα the quotient group (S ⊗ K) (H ⊗ L)/Imα, then β induces a homomorphism δ : Cokerα −→ H ⊗ K. On the other hand, the parts (iii) and (vi) of Lemma 3.1 show that Imα is invariant under the action of H ⊗ K and so, we have an action of H ⊗ K on Cokerα. We can hence get the following (ii) If I is a subgroup of Cokerα generated by the elements (x ⊗ y, (y ⊗ x)(∂(z) ⊗ z))Imα for all x, z ∈ S ∩ L, y ∈ H ∩ K, then (I, H K, δ) is a normal crossed submodule of (Cokerα, H ⊗ K, δ).
Without loss of generality, we may assume that v = (s ⊗ k, h ⊗ l)Imα and v 1 = (s 1 ⊗ k 1 , h 1 ⊗ l 1 )Imα. We must prove that or equivalently, letting The simple calculations, together with the results of Lemmas 2.2(i) and 3.1(i), allow us to get Putting a = ∂(s 2 l 3 ) ⊗ l 2 and b = ∂(l 3 ) ⊗ l 2 , we hence deduce that  where the right-hand side of the isomorphism denotes the tensor product of abelian crossed modules introduced in [19]. Also, one easily sees that the tensor and exterior products are symmetric.
The following extends [6; Proposition 2.3] and is important in obtaining the next results. Proof. (i) The only non-trivial part is to verify that τ 1 is a homomorphism. But this follows from the following observations. (1) τ 1 is well-defined, because it is induced by the homomorphism (S ⊗ K) (H ⊗ L) −→ T , (x, y) −→ λ S (x)λ L (y). (2) For any x, x ∈ S ⊗ K, y, y ∈ H ⊗ L, Note that the second equality follows from the first property of the crossed module λ S and the fact that λ L (y) ∈ L ∩ S.
(ii) It is sufficient to note that ker τ 2 = H K is a central subgroup of H ⊗ K and acts trivially on Cokerα.
(iii) We first assume that (L, K, ∂) is simply connected. Owing to part (ii), we only need to prove that ker τ 1 is contained in Cokerα H⊗K , or equivalently, x y = y for all x ∈ H ⊗ K and y ∈ ker τ 1 . Let x = h ⊗ k and y = (s ⊗ k, h ⊗ l)Imα for h, h ∈ H, k, k ∈ K, s ∈ S, l ∈ L. Then k ss −1 = h ll −1 and also, k = ∂(l ) for some l ∈ L, forcing that and using Lemma 3.1(i), It therefore follows that The proof for the other case is analogous.
The following corollary is an immediate consequence of the above proposition. In order to study the relation between the exterior product and tensor product of crossed modules, under some conditions, we use a generalized version of Whitehead's universal quadratic functor, the generalization being due to [19]. given by f (a 1 ⊗ a 2 ) = (∂(a 1 ) ⊗ a 2 , ∆(a 1 ⊗ a 2 ) −1 ), and ∂ Γ (b ⊗ a, γ(a 1 )) = ∆(b ⊗ ∂(a))γ(∂(a 1 )). Theorem 3.9. Let (T, G, ∂) be a crossed module such that ∂ is onto or G acts trivially on T . Then there is a natural exact sequence Proof. We only prove the result for the case of ∂ is onto. The proof for the other case is identical. It is sufficient to define a surjective morphism (η 1 , η 2 ) : (Γ((T, G, ∂) ab ), Γ(G ab ), ∂ Γ ) −→ (I, G G, δ). We take η 2 to be the epimorphism given in Proposition 2.1(vii). PuttingT = T /[G, T ] and G = G ab , we construct η 1 in the following three steps.
To prove (ii), it suffices to indicate that γ c+2 ((S, H, ∂) ⊗ (L, K, ∂)) = 1, or equivalently, by Lemma 4.1(ii), that ([ c+1 H ⊗ K, Cokerα], γ c+2 (H ⊗ K), δ) = 1. It follows from the assumption and the above extension that γ c+1 (H ⊗ K) ⊆ ker τ 2 and then γ c+2 (H ⊗ K) = 1. We now prove that [ c+1 H ⊗ K, Cokerα] = 1. To do this, we will first show by induction that for any n ≥ 1, every generator of [ n H ⊗ K, Cokerα] can be expressed as On the other hand, by the relation (2.5), It thus follows that Now, assume that the result holds for n ≥ 1. Then any generator of [ n+1 H ⊗ K, Cokerα] can be written as (y j ⊗ x j ) and applying arguments similar to the above, one can easily see that As an immediate consequence of the above theorem, we have In the following, we indicate a result similar to part (i) of the above corollary for nilpotent crossed modules.   A crossed module (T, G, ∂) is called nilpotent-by-finite (respectively, solvable-by-finite) if it has a nilpotent (respectively, solvable) normal crossed submodule (S, H, ∂) such that (T /S, G/H,∂) is finite.
In [9], it was established that if H and K are groups acting on each other compatibly, then H ⊗ K is nilpotent-by-finite or solvable-by-finite whenever H or K satisfy such information. In the final result of this section, we extend this result by showing the following (ii) if (S, H, ∂) or (L, K, ∂) is solvable-by-finite, then so is (S, H, ∂) ⊗ (L, K, ∂).
Proof. It is a routine exercise to check that the properties of nilpotent-byfinite and solvable-by-finite are closed under taking normal crossed submodules and central extensions. The results now follow from the extension (8) that is central because of Proposition 3.6(iii).