On the Second Homology of Crossed Modules

Abstract

In this article, we present a new description of the integral second homology of crossed modules of groups and generalize two basic results on the integral second homology of groups for crossed modules. Using these, we strengthen some consequences on covering pairs and the universal relative central extensions of pairs of finite groups.

1 Introduction

Several definitions for the homology of crossed modules have been given during the last years: Ellis [7] and Baues [2] introduced the homology of a crossed module to be the homology of its classifying space. Grandjeán and Ladra [10] defined the second homology crossed module by means of a Hopf formula applied to a particular kind of presentations called $\varepsilon$-projective. Also, associated with an extension of crossed modules, they [16] gave the construction of a five-term exact sequence for the homology of crossed modules. In continuation, Pirashvili [21] presented the notion of the tensor product of two abelian crossed modules, and he used it to construct the Ganea map, that is, extended the above five-term exact sequence one term further. Carrasco et al. [6], using the general theory of cotriple homology of Barr and Beck, defined the integral homology crossed modules of a crossed module as the simplicial derived functors of the abelianization functor from the category of crossed modules to the category of abelian crossed modules and generalized some classical results of the homology of groups. Considering projective presentations introduced in [6] instead of $\varepsilon$-projective presentations, they also obtained the results given in [10, 16].

Recently, authors [22] introduced the notions of the non-abelian tensor and exterior products of two normal crossed submodules of some crossed module of groups, which are generalizations of the works of Brown and Loday [4, 5] and Pirashvili [21].

In this article, we give a new description of the second homology of crossed modules; in fact, we describe the second homology of crossed modules as the central crossed submodules of their exterior products, which generalizes a result of Miller [18] for groups. Also, we show that the second homology of the direct product of two crossed modules is isomorphic to the direct product of the second homology of the factors and the tensor product of the two crossed modules abelianized. Finally, we give some applications to covering pairs and the universal central extensions of pairs of finite groups.

2 Preliminaries on Crossed Modules

In this section, we briefly recall some basic definitions in the category of crossed modules, which will be needed in the sequel.

A crossed module $\mathbf{T}=(T,G,\partial )$ is a group homomorphism $\partial :T\longrightarrow G$ together with an action of G on T, written $^{g}t$ for $t\in T$ and $g\in G$, satisfying $\partial (^gt)=g\partial (t)g^{-1}$ and $^{\partial (t)}t'=tt't^{-1}$, for all $t,t'\in T$, $g\in G$. It is worth noting that for any crossed module $(T,G,\partial )$, Im$\partial$ is a normal subgroup of G and $\ker \partial$ is a G-invariant subgroup in the center of T. Clearly, for any normal subgroup N of a group G, (N, G, i) is a crossed module, where i is the inclusion and G acts on N by conjugation. In this way, every group G can be seen as a crossed module in two obvious ways: (1, G, i) or (G, G, id).

A morphism of crossed modules $(\gamma _1,\gamma _2):(T,G,\partial )\longrightarrow (T',G',\partial ')$ is a pair of homomorphisms $\gamma _1:T\longrightarrow T'$ and $\gamma _2:G\longrightarrow G'$ such that $\partial '\gamma _1=\gamma _2\partial$ and $\gamma _1(^gt)=~^{\gamma _2(g)}\gamma _1(t)$ for all $g\in G$, $t\in T$.

Taking objects and morphisms as defined above, we obtain the category $\mathfrak {CM}$ of crossed modules. We refer the reader to [10, 20] for obtaining more information on this category.

Let $\mathbf{T}=(T,G,\partial )$ be a crossed module with normal crossed submodules $\mathbf{S}=(S,H,\partial )$ and $\mathbf{L}=(L,K,\partial )$. The following is a list of notations which will be used:

• $Z(\mathbf{T})=(T^G,Z(G)\cap st_G(T),\partial )$ is the center of $\mathbf{T}$, where Z(G) denotes the center of the group G, $T^G=\{t\in T|~^gt=t~\mathrm{for \ all}~g\in G\}$ and $st_G(T)=\{g\in G|~^gt=t~\mathrm{for \ all}~t\in T\}$.

• $\mathbf{T}'=([G,T],G',\partial )$ is the commutator crossed submodule of $\mathbf{T}$, where $G'=[G,G]$ and $[G,T]=\langle ~^gtt^{-1} \ | \ t\in T,g\in G\rangle$ is the displacement subgroup of T relative to G.

• $[\mathbf{S},\mathbf{L}]$ is the normal crossed submodule $([K,S][H,L],[H,K],\partial )$ of $\mathbf{T}$.

• $\mathbf{T}_{ab}=(T/[G,T],G_{ab},\overline{\partial })$ is the abelianization of $\mathbf{T}$, where $G_{ab}=G/G'$ and $\overline{\partial }$ is induced by $\partial$.

A crossed module $(T,G,\partial )$ is perfect if it coincides with its commutator crossed submodule and is abelian if it coincides with its center.

In [6], it is proved that the category of crossed modules is an algebraic category, that is, there is a tripleable forgetful functor from the category $\mathfrak {CM}$ to the category of sets $\mathfrak {Set}$ and is deduced that every crossed module admits a projective presentation. Also, it is shown in [1] that if $(Y,F,\mu )$ is a projective crossed module, then it can be assumed that $\mu$ is inclusion and the groups F, F/Y are free objects in the category of groups. Now, let $(V,R,\mu )\rightarrowtail (Y,F,\mu )\twoheadrightarrow (T,G,\partial )$ be a projective presentation of the crossed module $(T,G,\partial )$. It is proved in [6] that the second homology crossed module of $(T,G,\partial )$ is, up to isomorphism, the abelian crossed module

$$\begin{aligned} \left( \frac{V\cap [F,Y]}{[R,Y][F,V]},\frac{R\cap F'}{[F,R]},\bar{\mu }\right) . \end{aligned}$$

Considering a group G as a crossed module in the two usual ways, we have $H_2(1,G,i)=(1,H_2(G),i)$ or $H_2(G,G,id)=(H_2(G),H_2(G),id)$, which gives the Hopf’s formula [13]. Note that in the above projective presentation of $(T,G,\partial )$, if $\partial$ is injective, then $V=R\cap Y$.

A central extension $e:(A,B,\partial ^*)\rightarrowtail (T^*,G^*,\partial ^*)\twoheadrightarrow (T,G,\partial )$ is called a stem extension of $(T,G,\partial )$ if $(A,B,\partial ^*)\subseteq (T^*,G^*,\partial ^*)'$. If, in addition, $(A,B,\partial ^*)\cong H_2(T,G,\partial )$, then e is called a stem cover. In this case, $(T^*,G^*,\partial ^*)$ is said to be a covering crossed module of $(T,G,\partial )$. In [19], it is proved that any crossed module $(T,G,\partial )$ admits at least one covering crossed module and is determined the structure of all stem covers of a crossed module whose second homology is finite.

Finally, we recall from [22] that the non-abelian tensor and exterior products of two normal crossed submodules $(S,H,\partial )$ and $(L,K,\partial )$ of a given crossed module are defined, respectively, as

$$\begin{aligned}&(S,H,\partial )\otimes (L,K,\partial )=~(\mathrm{coker}\alpha ,H\otimes K,\delta ),\\&(S,H,\partial )\wedge (L,K,\partial )=\left( \frac{\mathrm{coker}\alpha }{I},H\wedge K,\bar{\delta }\right) , \end{aligned}$$

in which $\alpha =(id_S\otimes \partial ,(\partial \otimes id_L)^{-1})$ and the map $\delta$ is induced on $\mathrm{coker}\alpha$ by the homomorphism $\beta =\langle \partial \otimes id_K,id_H\otimes \partial \rangle$:

Also, I is a normal subgroup of $\mathrm{coker}\alpha$ generated by the elements $(x\otimes y,(y\otimes x)(\partial (z)\otimes z))$Im$\alpha$ for all $x,z\in S\cap L$, $y\in H\cap K$. In the case of abelian crossed modules, the definition of the tensor product holds for any two abelian crossed modules (see [21]).

The proof of the following lemma is straightforward, and it will be left to the reader.

Lemma 2.1

Let $\mathbf{A}_{\mathbf{1}}=(A_1,B_1,\partial _1)$, $\mathbf{A}_{\mathbf{2}}=(A_2,B_2,\partial _2)$ and $\mathbf{A}=(A,B,\partial )$ be abelian crossed modules. Then

(i) If $f=(f_1,f_2):\mathbf{A}_{\mathbf{1}}\longrightarrow \mathbf{A}$ and $g=(g_1,g_2):\mathbf{A}_{\mathbf{2}}\longrightarrow \mathbf{A}$ are morphisms of crossed modules, then $f*g=(f_1*g_1, f_2*g_2):\mathbf{A}_{\mathbf{1}}\times \mathbf{A}_{\mathbf{2}} \longrightarrow \mathbf{A}$ is a morphism of crossed modules, where $f_1*g_1(a_1,a_2)=f_1(a_1)g_1(a_2)$ and $f_2*g_2(b_1,b_2)=f_2(b_1)g_2(b_2)$, for all $a_i \in A_i, b_i \in B_i, i=1,2$.

(ii) There exists an isomorphism $\nu =(\nu _1,\nu _2):\mathbf{A}_{\mathbf{1}}\otimes \mathbf{A}_{\mathbf{2}}\longrightarrow \mathbf{A}_{\mathbf{2}}\otimes \mathbf{A}_{\mathbf{1}}$ defined by

$$\begin{aligned}&\nu _1((a_1\otimes b_2,b_1\otimes a_2)Im\alpha )\\&\quad =\big ((a_2\otimes b_1)^{-1},(b_2\otimes a_1)^{-1}\big )Im\alpha '~~\mathrm{and}~~\nu _2(b_1\otimes b_2)=(b_2\otimes b_1)^{-1}. \end{aligned}$$

The next proposition provides some properties of the above notions, which is useful in our investigation.

Proposition 2.2

[22] Let $\mathbf{T}=(T,G,\partial )$ be a crossed module. Then

(i) There is a morphism of crossed modules $(\tau _1,\tau _2){:}\,\mathbf{T}\otimes \mathbf{T}\longrightarrow \mathbf{T}$, where $Im(\tau _1,\tau _2)=\mathbf{T}'$ and $J_2(\mathbf{T}):=\ker (\tau _1,\tau _2)$ is abelian.

(ii) There is an isomorphism of crossed modules $(\widetilde{\varphi },id){:}\,\mathbf{T}\wedge \mathbf{T}\longrightarrow (G\wedge T,G\wedge G,id\wedge \partial )$, where $\widetilde{\varphi }$ is induced by $\mu :(T\otimes G)\rtimes (G\otimes T)\longrightarrow G\otimes T$, $\mu (x,y)=\theta (x)y$, in which $\theta {:}\,T\otimes G \longrightarrow G\otimes T$ is defined by $\theta (t\otimes g)=(g\otimes t)^{-1}$.

(iii) Let $\mathbf{S}$ and $\mathbf{L}$ be normal crossed submodules of $\mathbf{T}$. If $[\mathbf{S},\mathbf{L}]=1$, then $\mathbf{S}\otimes \mathbf{L}\cong \mathbf{S}_{ab}\otimes \mathbf{L}_{ab}$.

(iv) If T is simply connected (that is, $\partial$ is onto), then there is a natural exact sequence

$$\begin{aligned} \Gamma (\mathbf{T}_{ab})\longrightarrow \mathbf{T}\otimes \mathbf{T} \twoheadrightarrow \mathbf{T}\wedge \mathbf{T}, \end{aligned}$$

in which $\Gamma (\mathbf{T}_{ab})$ is a generalized version of Whitehead’s universal quadratic functor (see [21] for more information).

3 Main Results

Miller [18] proves that, for any group G, the second homology of G is isomorphic to the kernel of the commutator map $G\wedge G{\mathop {\longrightarrow }\limits ^{[~,~]}}G$. Using this result, he determines the behavior of the functor $H_2(-)$ with respect to the direct product of groups and, also, Ellis [8] gets a six-term exact sequence in homology

$$\begin{aligned} H_2(G,N)\longrightarrow H_2(G)\longrightarrow H_2(Q)\longrightarrow N/[G,N]\longrightarrow H_1(G) \twoheadrightarrow H_1(Q), \end{aligned}$$

from a short exact sequence of groups $N\rightarrowtail G\twoheadrightarrow Q$. Here, $H_2(G,N)$ denotes the second relative Chevally–Eilenberg homology of the pair (G, N), which is isomorphic to $\ker (G\wedge N\longrightarrow G)$ (see [5]). In this section, we generalize these results to crossed modules. In fact, we prove

Theorem 3.1

(i) For arbitrary crossed module $\mathbf{T}=(T,G,\partial )$, $H_2(\mathbf{T})\cong \ker (\mathbf{T}\wedge \mathbf{T}\longrightarrow \mathbf{T})$. In particular, $H_2(\mathbf{T})\cong (\ker (G\wedge T\longrightarrow T),\ker (G\wedge G\longrightarrow G),id\wedge \partial )$.

(ii) If $e: \mathbf{S}\rightarrowtail \mathbf{T}\twoheadrightarrow \mathbf{M}$ is an extension of crossed modules, then there is a natural exact sequence

$$\begin{aligned} \ker (\mathbf{T}\wedge \mathbf{S}\longrightarrow \mathbf{T})\longrightarrow H_2(\mathbf{T})\longrightarrow H_2(\mathbf{M})\longrightarrow \frac{\mathbf{S}}{[\mathbf{S},\mathbf{T}]}\longrightarrow H_1(\mathbf{T})\twoheadrightarrow H_1(\mathbf{M}), \end{aligned}$$

where $H_1(-)$ denotes the first homology of crossed modules [11].

Theorem 3.2

For crossed modules $\mathbf{T}_{\mathbf{1}}=(T_1,G_1,\partial _1)$ and $\mathbf{T}_{\mathbf{2}}=(T_2,G_2,\partial _2)$, there are isomorphisms

$$\begin{aligned} J_2(\mathbf{T}_{\mathbf{1}}\times \mathbf{T}_{\mathbf{2}})&\cong J_2(\mathbf{T}_{\mathbf{1}})\times J_2(\mathbf{T}_{\mathbf{2}})\times (\mathbf{T}_{{\mathbf{1}}_{ab}}\otimes \mathbf{T}_{{\mathbf{2}}_{ab}})\times (\mathbf{T}_{{\mathbf{2}}_{ab}}\otimes \mathbf{T}_{{\mathbf{1}}_{ab}}),\\ H_2(\mathbf{T}_{\mathbf{1}}\times \mathbf{T}_{\mathbf{2}})&\cong H_2(\mathbf{T}_{\mathbf{1}})\times H_2(\mathbf{T}_{\mathbf{2}})\times (\mathbf{T}_{{\mathbf{1}}_{ab}}\otimes \mathbf{T}_{{\mathbf{2}}_{ab}}). \end{aligned}$$

In order to prove these theorems, we need the following two propositions, which are generalizations of [8, Proposition 1] and [3, Proposition 10], respectively.

Proposition 3.3

Let $\mathbf{S}=(S,H,\partial )$ be a normal crossed submodule of a crossed module $\mathbf{T}=(T,G,\partial )$. Then there is an exact sequence of crossed modules

Proof

By the definition of the exterior product of crossed modules, we set

$$\begin{aligned}&\mathbf{T}\wedge \mathbf{S}=(\mathrm{coker}\alpha _1/I_1,G\wedge H,\bar{\delta }_1)~,~ \mathbf{T}\wedge \mathbf{T}=(\mathrm{coker}\alpha /I,G\wedge G,\bar{\delta }),\\&\quad ~\frac{\mathbf{T}}{\mathbf{S}}\wedge \frac{\mathbf{T}}{\mathbf{S}}=(\mathrm{coker}\bar{\alpha }/ \bar{I},\frac{G}{H}\wedge \frac{G}{H},\widetilde{\delta }). \end{aligned}$$

It is easy to see that the natural epimorphisms $T\longrightarrow T/S$ and $G\longrightarrow G/H$ induce the homomorphisms $\pi _1:\mathrm{coker}\alpha /I\longrightarrow \mathrm{coker}\bar{\alpha }/\bar{I}$ and $\pi _2: G\wedge G\longrightarrow G/H\wedge G/H$ such that $\pi =(\pi _1,\pi _2)$ is a surjective morphism. Also, the functional homomorphisms $\beta _1: T\otimes H\longrightarrow T\otimes G$ and $\beta _2: G\otimes S\longrightarrow G\otimes T$ give rise to the homomorphism $\varphi : (T\otimes H)\rtimes (G\otimes S)\longrightarrow (T\otimes G)\rtimes (G\otimes T)$ defined by $\varphi (\nu ,\omega )=(\beta _1(\nu ),\beta _2(\omega ))$. Since $\varphi (Im\alpha _1)\subseteq Im\alpha$, we can obtain the homomorphism $\widetilde{\varphi }:\mathrm{coker}\alpha _1\longrightarrow \mathrm{coker}\alpha$ induced by $\varphi$. Certainly, $\widetilde{\varphi }(I_1)\subseteq I$ and so $\widetilde{\varphi }$ induces a homomorphism $\eta _1:\mathrm{coker}\alpha _1/I_1\longrightarrow \mathrm{coker}\alpha /I$. It is straightforward that $\eta =(\eta _1,\eta _2)$ is a morphism of crossed modules in which $\eta _2:G\wedge H\longrightarrow G\wedge G$ is the functional homomorphism. To complete the proof, we need to indicate that Im$\eta =\ker \pi$. We have Im$\eta _2=\ker \pi _2$, thanks to Ellis [8, Proposition 1]. It is easily verified that Im$\eta _1$ is a normal subgroup of $\mathrm{coker}\alpha /I$ contained in $\ker \pi _1$. We now show that this inclusion is an equality by constructing an isomorphism $\widetilde{\kappa }:\mathrm{coker}\bar{\alpha }/\bar{I}\longrightarrow \mathrm{coker}\eta _1$. Consider the maps $e_1: T/S\times G/H\longrightarrow \mathrm{coker}\eta _1$ and $e_2: G/H\times T/S\longrightarrow \mathrm{coker}\eta _1$ defined by $e_1(tS,gH)=\overline{(t\otimes g,1)}Im\eta _1$ and $e_2(gH,tS)=\overline{(1,g\otimes t)}$Im$\eta _1$, respectively. For $t_1, t_2\in T$, $g_1,g_2\in G$, $s\in S$, $h\in H$, if $t_1=t_2s$, $g_1=g_2h$, then

$$\begin{aligned} e_1(t_1S,g_1H)&=\overline{(t_2s\otimes g_2h,1)}Im\eta _1\\&=\overline{(s\otimes g_2,1)}\overline{(s\otimes h,1)}\overline{(t_2\otimes g_2,1)}\overline{(t_2\otimes h,1)}Im\eta _1\\&=\overline{(s\otimes g_2,1)}\overline{(t_2\otimes g_2,1)}Im\eta _1, \end{aligned}$$

because $\overline{(s\otimes h,1)}, \overline{(t_2\otimes h,1)}\in$ Im$\eta _1$. On the other hand, we have $\overline{(s\otimes g_2,1)}=\overline{(1,g_2\otimes s)}^{-1}\in Im\eta _1$, since $\overline{(s\otimes g_2,g_2\otimes s)}=1$ in $\mathrm{coker}\alpha /I$. It is therefore inferred that $e_1(t_1S,g_1H)=\overline{(t_2\otimes g_2,1)}Im\eta _1 =e_1(t_2S,g_2H)$. So, $e_1$ and, similarly, $e_2$ are well-defined. It is easy to verify that $e_1$ and $e_2$ are crossed pairings, and the universal property of the tensor product thus yields the homomorphisms $\bar{e}_1: T/S\otimes G/H\longrightarrow \mathrm{coker}\eta _1$ and $\bar{e}_2: G/H\otimes T/S\longrightarrow \mathrm{coker}\eta _1$. Now, the map

$$\begin{aligned} \kappa :(T/S\otimes G/H)\rtimes (G/H\otimes T/S)\longrightarrow \mathrm{coker}\eta _1 \end{aligned}$$

defined by $\kappa (x,y)=\bar{e}_1(x)\bar{e}_2(y)$ is a homomorphism that annihilates $Im\bar{\alpha }$. So $\kappa$ induces the homomorphism $\bar{\kappa }: \mathrm{coker}\bar{\alpha }\longrightarrow \mathrm{coker}\eta _1$ with $\bar{\kappa }(\bar{I})=0$, which in turn induces a homomorphism $\widetilde{\kappa }: \mathrm{coker}\bar{\alpha }/\bar{I}\longrightarrow \mathrm{coker}\eta _1$. It is routine to check that $\widetilde{\kappa }$ is an isomorphism with inverse induced by $\pi _2$. The proof is complete. $\square$

Proposition 3.4

Let $\mathbf{M}=(M,P,\partial _1)$ and $\mathbf{N}=(N,Q,\partial _2)$ be two crossed modules and $\mathbf{T}=(T,G,\partial _1\times \partial _2)$ be a normal crossed submodule of $\mathbf{M}\times \mathbf{N}$. Then

$$\begin{aligned} (\mathbf{M}\times \mathbf{N})\otimes \mathbf{T}\cong (\mathbf{M}\otimes \mathbf{T})\times (\mathbf{N}\otimes \mathbf{T}). \end{aligned}$$

Proof

Using the definition of the tensor product of crossed modules, we suppose that $\mathbf{M}\otimes \mathbf{T}=(\mathrm{coker}\alpha _1,P\otimes G,\delta _1),\mathbf{N}\otimes \mathbf{T}=(\mathrm{coker}\alpha _2,Q\otimes G,\delta _2)$ and $(\mathbf{M}\times \mathbf{N})\otimes \mathbf{T}=(\mathrm{coker}\alpha ,(P\times Q)\otimes G,\delta )$, where $\alpha _1=(id_M\otimes (\partial _1\times \partial _2),(\partial _1\otimes id_T)^{-1})$, $\alpha _2=(id_N\otimes (\partial _1\times \partial _2),(\partial _2\otimes id_T)^{-1})$ and $\alpha =(id_{M\times N}\otimes (\partial _1\times \partial _2),((\partial _1\times \partial _2)\otimes id_T)^{-1})$. We only need to define an isomorphism $(\psi _1,\psi _2)$

The second component $\psi _2$ is the isomorphism given in [3, Proposition 10], which is defined on generators by $\psi _2((p,q)\otimes g)=(p\otimes g,q\otimes g)$. We now construct $\psi _1$, which will be induced on $\mathrm{coker}\alpha$ by a homomorphism $\langle \phi _1,\phi _2\rangle$

Let us define $\phi _1$ and $\phi _2$ on generators as follows:

$$\begin{aligned} \phi _1((m,n)\otimes g)= & {} ((m\otimes g,1)\mathrm{Im}\alpha _1,(n\otimes g,1)\mathrm{Im}\alpha _2),\\ \phi _2((p,q)\otimes t)= & {} ((1,p\otimes t)\mathrm{Im}\alpha _1,(1,q\otimes t)\mathrm{Im}\alpha _2). \end{aligned}$$

Since the action of M on groups N and Q is trivial, for all $m\in M$, $n\in N$ and $g=(g_1,g_2)\in G$, we have

$$\begin{aligned} ^m(n\otimes g)&=\big (m\otimes ~^ngg^{-1}\big )(n\otimes g)= \big (m\otimes ~^{(1,n)}(g_1,g_2)\big (g_1^{-1},g_2^{-1}\big )\big )(n\otimes g)\\&=\big (m\otimes ~^ng_2g_2^{-1}\big )(n\otimes g)= ~^m(n\otimes g_2)(n\otimes g_2)^{-1}(n\otimes g)=n\otimes g. \end{aligned}$$

So, M on $N\otimes G$ and similarly, N on $M\otimes G$ act trivially. Using these results and after a long calculations, one can see that $\phi _1$ and $\phi _2$ preserve the defining relations of the tensor product of groups and are then homomorphisms. We now claim that $\phi _1(^ba)=~^{\phi _2(b)}\phi _1(a)$ for all $a\in (M\times N)\otimes G$ and $b\in (P\times Q)\otimes T$. Without loss of generality, we may assume that $a=(m,n)\otimes g$ and $b=(p,q)\otimes (t_1,t_2)$, where $(t_1,t_2)\in T$. Then

$$\begin{aligned} \phi _1(^ba)&=\phi _1\big (^{(p,q)\otimes (t_1,t_2)}((m,n)\otimes g)\big )= \phi _1\big (^{(^pt_1t_1^{-1},~^qt_2t_2^{-1})}((m,n)\otimes g)\big )\\&=(\overline{(^{^p{t_1}{t_1}^{-1}}m\otimes ~^{(^pt_1t_1^{-1},^qt_2t_2^{-1})}g,1)}, \overline{(^{^qt_2t_2^{-1}}n\otimes ~^{(^pt_1t_1^{-1},^qt_2t_2^{-1})}g,1)})\\&=(\overline{(^{^qt_2t_2^{-1}}(^{^pt_1t_1^{-1}}(m\otimes g)),1)}, \overline{(^{^pt_1t_1^{-1}}(^{^qt_2t_2^{-1}}(n\otimes g)),1)})\\&=(\overline{(^{p\otimes t}(m\otimes g),1)}, \overline{(^{q\otimes t}(n\otimes g),1)})\\&=^{(\overline{(1,p\otimes t)},\overline{(1,q\otimes t)})} (\overline{(m\otimes g,1)},\overline{(n\otimes g,1)})\\&=~^{\phi _2((p,q)\otimes t)}\phi _1((m,n)\otimes g). \end{aligned}$$

Note that the forth equality follows from the fact that N acts trivially on $M\otimes G$. It, therefore, follows from Lemma 2.1(i) that the map $\phi =\langle \phi _1,\phi _2\rangle :((M\times N)\otimes G)\rtimes ((P\times Q)\otimes T)\longrightarrow \mathrm{coker}\alpha _1\times \mathrm{coker}\alpha _2$ defined by $\phi (a,b)=(\phi _1(a),\phi _2(b))$ is a homomorphism. Because $\phi$ annihilates Im$\alpha$, $\phi$ induces the homomorphism $\psi _1$. We prove that $\psi _1$ is an isomorphism by giving an inverse for it. Consider the canonical homomorphisms $\eta _1:\mathrm{coker}\alpha _1\longrightarrow \mathrm{coker}\alpha$, $(m\otimes g,p\otimes t)$Im$\alpha _1\longmapsto ((m,1)\otimes g,(p,1)\otimes t)$Im$\alpha$, and $\eta _2:\mathrm{coker}\alpha _2\longrightarrow \mathrm{coker}\alpha$, $(n\otimes g,q\otimes t)$Im$\alpha _2\longmapsto ((1,n)\otimes g,(1,q)\otimes t)$Im$\alpha$. Then the map $\eta =\langle \eta _1,\eta _2\rangle : \mathrm{coker}\alpha _1\times \mathrm{coker}\alpha _2\longrightarrow \mathrm{coker}\alpha$ given by $\eta (x,y)=\eta _1(x)\eta _2(y)$ is an inverse for $\psi _1$. One easily sees that $(\psi _1,\psi _2)$ is a morphism of crossed modules. $\square$

Proof of Theorem 3.1

(i) Let $(V,R,\mu )\rightarrowtail (Y,F,\mu ) \twoheadrightarrow (T,G,\partial )$ be a projective presentation of the crossed module $(T,G,\partial )$. It is sufficient to prove

$$\begin{aligned} (T,G,\partial )\wedge (T,G,\partial )&\cong \frac{(Y,F,\mu )'}{[(Y,F,\mu ),(V,R,\mu )]}. \end{aligned}$$

Since $(Y,F,\mu )$ is a projective crossed module, F/Y is a free group. It follows from [8, Theorem 6] that the kernel of the epimorphism $\lambda _Y:F\wedge Y\longrightarrow [F,Y]$ is trivial. Also, the homomorphism $\lambda _F:F\wedge F\longrightarrow F'$ is an isomorphism thanks to Ellis [8, Proposition 2]. Now, it is readily seen that $(\lambda _Y, \lambda _F ):(F\wedge Y, F\wedge F, id\wedge \mu )\longrightarrow ([F,Y],F',\mu )$ is an isomorphism of crossed modules and

$$\begin{aligned} (Y,F,\mu )\wedge (Y,F,\mu )\cong ([F,Y],F',\mu ). \end{aligned}$$

We consider the diagram

where $(\eta _1,\eta _2)$ and $(\widetilde{\varphi },id)$ are the morphisms given in Propositions 2.2(ii) and 3.3, respectively, and $\psi =(\psi _1,\psi _2)$ is the composition of $(\widetilde{\varphi },id)$ and $(\eta _1,\eta _2)$. As Im$\psi _1=\langle f\wedge \nu ,r\wedge y \ |\ f\in F,\nu \in V,r\in R,y\in Y\rangle$ and Im$\psi _2=\langle r\wedge f \ | \ r\in R,f\in F\rangle$, for the morphism $(\lambda _Y,\lambda _F)$ given above, we have

$$\begin{aligned} (\lambda _Y,\lambda _F)(\mathrm{Im}\psi _1,\mathrm{Im}\psi _2,id\wedge \mu ) =([F,V][R,Y],[R,F],\mu ), \end{aligned}$$

and therefore,

$$\begin{aligned} (T,G,\partial )\wedge (T,G,\partial )\cong \frac{(F\wedge Y,F\wedge F,id\wedge \mu )}{\mathrm{Im}\psi }\cong \frac{(Y,F,\mu )'}{[(Y,F,\mu ),(V,R,\mu )]}. \end{aligned}$$

The proof is complete.

(ii) There is the following commutative diagram of crossed modules:

where the rows and, invoking Proposition 3.3, columns are exact. Then, by the snake lemma, we get the following exact sequence

$$\begin{aligned} \ker (\mathbf {T}\wedge \mathbf {S}\longrightarrow \mathbf {T})\longrightarrow \ker (\mathbf {T}\wedge \mathbf {T}\longrightarrow \mathbf {T})\longrightarrow \ker (\mathbf{M}\wedge \mathbf{M}\longrightarrow \mathbf{M}), \end{aligned}$$

and the result now follows from part (i) and [6, Theorem 12(i)]. $\square$

Proof of Theorem 3.2

By Proposition 2.2(i) and Theorem 3.1(i), we only need to prove that

$$\begin{aligned} (\mathbf{T_1}\times \mathbf{T_2})\otimes (\mathbf{T_1}\times \mathbf{T_2})\cong & {} (\mathbf{T_1}\otimes \mathbf{T_1})\times (\mathbf{T_2}\otimes \mathbf{T_2})\times (\mathbf{T_1}_{ab}\nonumber \\&\otimes \mathbf{T_2}_{ab})\times (\mathbf{T_2}_{ab}\otimes \mathbf{T_1}_{ab}), \end{aligned}$$

(1)

$$\begin{aligned} (\mathbf{T_1}\times \mathbf{T_2}) \wedge (\mathbf{T_1}\times \mathbf{T_2})\cong & {} (\mathbf{T_1}\wedge \mathbf{T_1})\times (\mathbf{T_2}\wedge \mathbf{T_2})\times (\mathbf{T_1}_{ab}\otimes \mathbf{T_2}_{ab}). \end{aligned}$$

(2)

Identifying $\mathbf{T_1}$ and $\mathbf{T_2}$ with their images in the crossed module $\mathbf{T_1}\times \mathbf{T_2}$, $[\mathbf{T_1},\mathbf{T_2}]=1$ and so, according to Proposition 2.2(iii), $\mathbf{T_1}\otimes \mathbf{T_2}\cong \mathbf{T_1}_{ab}\otimes \mathbf{T_2}_{ab}$ and $\mathbf{T_2}\otimes \mathbf{T_1}\cong \mathbf{T_2}_{ab}\otimes \mathbf{T_1}_{ab}$. We therefore obtain the isomorphism (1) by applying Proposition 3.4 twice.

To prove the isomorphism (2), we consider the following diagram

where $\psi$, $\varphi$ are isomorphisms obtained in the proof of Proposition 3.4, i is the inclusion morphism, and $id_{\mathbf{T_1}_{ab}\otimes \mathbf{T_2}_{ab}}*\nu :(\mathbf{T_1}_{ab}\otimes \mathbf{T_2}_{ab})\times (\mathbf{T_2}_{ab}\otimes \mathbf{T_1}_{ab})\longrightarrow \mathbf{T_1}_{ab}\otimes \mathbf{T_2}_{ab}$ is the surjective morphism given in Lemma 2.1(i). An easy calculation shows that

$$\begin{aligned}&\lambda \varphi ((\mathbf{T}_1\times \mathbf{T}_2)\square (\mathbf{T}_1\times \mathbf{T}_2)) \subseteq (\mathbf{T}_1\square \mathbf{T}_1)\times (\mathbf{T}_2\square \mathbf{T}_2),\\&\quad \psi i((\mathbf{T}_1\square \mathbf{T}_1)\times (\mathbf{T}_2\square \mathbf{T}_2)) \subseteq (\mathbf{T}_1\times \mathbf{T}_2)\square (\mathbf{T}_1\times \mathbf{T}_2), \end{aligned}$$

from which we get the induced morphisms $\eta$ and $\theta$. Since $\eta$ is surjective, we prove that $\eta$ is isomorphism by showing that $\theta$ is a left inverse of $\eta$. But this follows from the following observations. For any $t_i\in T_i$, $g_i\in G_i$ ($i=1,2$), we have

$$\begin{aligned} (t_1,1)\otimes (g_1,g_2)= & {} ((t_1,1)\otimes (1,g_2))^{(1,g_1)}{((t_1,1)\otimes (g_1,1))}\\= & {} ((t_1,1)\otimes (1,g_2))((t_1,1)\otimes (g_1,1)), \end{aligned}$$

and similarly, $(g_1,1)\otimes (t_1,t_2)=((g_1,1)\otimes (t_1,1))((g_1,1)\otimes (1,t_2)$. The proof is complete. $\square$

We have the following consequences of the above theorems.

Corollary 3.5

Let (N, G, i) be the inclusion crossed module. Then $\displaystyle H_2(G,N)\cong \frac{R\cap [F,Y]}{[R,Y][F,R\cap Y]}$, where $(R\cap Y,R,\mu ) \rightarrowtail (Y,F,\mu )\twoheadrightarrow (N,G,i)$ is a projective presentation of (N, G, i).

Proof

It follows from Theorem 3.1(i) and Hopf formula for $H_2(N,G,i)$. $\square$

Corollary 3.6

Let $(N_1,G_1,i_1)$ and $(N_2,G_2,i_2)$ be inclusion crossed modules. Then

$$\begin{aligned} H_2(G_1\times G_2,N_1\times N_2)&\cong H_2(G_1,N_1)\times H_2(G_2,N_2)\\&\quad \times \left( \frac{(\overline{N}_1\otimes \overline{G}_2) \times (\overline{G}_1\otimes \overline{N}_2)}{\langle (\overline{n_1}\otimes \overline{i_2(n_2)}, (\overline{i_1(n_1)}\otimes \overline{n_2})^{-1}) |n_1\in N_1,n_2\in N_2\rangle }\right) ,\\ H_2(G_1\times G_2)&\cong H_2(G_1)\times H_2(G_2)\times (\overline{G}_1\otimes \overline{G}_2). \end{aligned}$$

Proof

It follows from Theorem 3.2 and its proof. $\square$

Corollary 3.7

For any simply connected $\mathbf{T}=(T,G,\partial )$, there is a commutative diagram of crossed modules with exact rows and columns.

Proof

It follows from Proposition 2.2(iv) and Theorem 3.1(i). $\square$

Corollary 3.8

For any finite perfect crossed module $\mathbf{T}=(T,G,\partial )$, the second homology of its covers is trivial.

Proof

In view of [20, Theorem 2.68], the extension

$$\begin{aligned} \ker (\tau _1,\tau _2)\rightarrowtail (G\otimes T,G\otimes G,id\otimes \partial ){\mathop {\twoheadrightarrow }\limits ^{(\tau _1,\tau _2)}}\mathbf{T} \end{aligned}$$

(3)

is the universal central extension of T, in which the commutator morphism $(\tau _1,\tau _2)$ is given by $\tau _1(g_1\otimes t)=~^{g_1}tt^{-1}$ and $\tau _2(g_1\otimes g_2)=[g_1,g_2]$, for all $t\in T$, $g_1,g_2\in G$. Due to [24, Proposition 5 and Corollary 2], the extension (3) is, up to isomorphism, the only stem cover of T. Applying Theorem 3.1(ii) to the extension (3), we deduce that $H_2(G\otimes T,G\otimes G,id\otimes \partial )$ is a homomorphic image of $(G\otimes T,G\otimes G,id\otimes \partial )\wedge \ker (\tau _1,\tau _2)$. But the perfectness of the crossed module T, together with Proposition 2.1(iii) yields that

$$\begin{aligned} (G\otimes T,G\otimes G,id\otimes \partial )\wedge \ker (\tau _1,\tau _2)=(G\otimes T,G\otimes G,id\otimes \partial )\otimes \ker (\tau _1,\tau _2)=1, \end{aligned}$$

and then the result holds. $\square$

4 Applications to the Pair of Groups

Loday [17] and Ellis [9], respectively, introduced the notions of relative central extension and cover for pairs of groups and used them as useful tools to develop the theory of capability, Schur multiplier, and central series of groups to a theory for pairs of groups. This section is devoted to structural results on these notions.

Let (G, N) be a pair of groups, in which N is a normal subgroup of G. We recall from [9, 17] that

• A crossed module $(M,G,\delta )$ is called a relative central extension of (G, N) if $\delta (M)=N$ and $\ker \delta \subseteq M^G$. A morphism between two relative central extensions $\mathbf{M}=(M,G,\delta )$ and $\mathbf{M'}=(M',G,\delta ')$ of (G, N) is a crossed module morphism $(f,id):\mathbf{M}\longrightarrow \mathbf{M'}$. In particular, if f is surjective, then $\mathbf{M'}$ is called a homomorphic image of M.

• A relative central extension $(M,G,\delta )$ of the pair (G, N) is called universal if there exists a unique morphism from it to any relative central extension of (G, N). It is proved in [17] that the pair (G, N) has a universal relative central extension if and only if N is G-perfect (that is, $[G,N]=N$).

• A relative central extension $(M,G,\delta )$ of (G, N) is called a relative stem extension if $\ker \delta \subseteq [G,M]$. If, in addition, $ker\delta \cong H_2(G,N)$ then $(M,G,\delta )$ is said to be a covering pair for (G, N). It is established in [19] that any pair of groups admits at least one covering pair.

We begin with the following key lemma, which deals with the connection between the stem covers and the universal central extensions of crossed modules with corresponding concepts for pairs of groups.

Lemma 4.1

Let (G, N) be a pair of groups and $e{:}\, (A,B,\delta )\rightarrowtail (M,H,\delta ){\mathop {\twoheadrightarrow }\limits ^{(\psi _1,\psi _2)}}(N,G,i)$ be a stem cover (respectively, universal central extension) of the inclusion crossed module (N, G, i). Then the composite homomorphism $i\psi _1: M\longrightarrow G$ with the action of G on M given by $^gm=^{h}m$, in which h is any element in the pre-image of g via $\psi _2$, is a covering pair (respectively, universal relative central extension) of (G, N).

Proof

The case of stem cover follows from [19, Corollary 3.2]. We so assume that e is a universal central extension of (N, G, i), and $(M_1,G,\delta _1)$ is an arbitrary relative central extension of the pair (G, N). Considering the central extension $(ker\delta _1,1,1)\rightarrowtail (M_1,G,\delta _1){\mathop {\twoheadrightarrow }\limits ^{(\delta _1,id)}}(N,G,i)$, we get a unique morphism $(\beta _1,\psi _2):(M,H,\delta )\longrightarrow (M_1,G,\delta _1)$ such that $(\delta _1,id)(\beta _1,\psi _2)=(\psi _1,\psi _2)$. It is straightforward to see that $(\beta _1,id):(M,G,i\psi _1)\longrightarrow (M_1,G,\delta _1)$ is a morphism of crossed modules. This morphism is unique, because if $(\beta _1',id):(M,G,i\psi _1)\longrightarrow (M_1,G,\delta _1)$ is another morphism, then $(\beta _1',\psi _2):(M,H,\delta )\longrightarrow (M_1,G,\delta _1)$ is a morphism satisfying $(\delta _1,id)(\beta _1',\psi _2)=(\psi _1,\psi _2)$. But this result yields that $\beta _1=\beta _1'$, as desired. $\square$

The following corollary is a consequence of the above lemma.

Corollary 4.2

Let $(R\cap Y,R,\mu )\rightarrowtail (Y,F,\mu ){\mathop {\twoheadrightarrow }\limits ^{(\pi _1,\pi _2)}}(N,G,i)$ be a projective presentation of a perfect inclusion crossed module (N, G, i). Then $([F,Y]/([R,Y][F,R\cap Y],G,i\bar{\pi _1})$ is a covering pair as well as the universal relative central extension of the pair (G, N), where $\bar{\pi _1}$ is induced by $\pi _1$.

Proof

This follows from Lemma 4.1, [23, Proposition 4.3] and [6, Page 171]. $\square$

In the following theorem, we see how Theorem 3.1(i) together with [19, Theorem 3.6] can be used to determine the structure of covering pairs.

Theorem 4.3

Let (G, N) be a pair of finite groups, and $(R\cap Y,R,\mu )\rightarrowtail (Y,F,\mu )\twoheadrightarrow (N,G,i)$ be a projective presentation of the crossed module (N, G, i). Then the crossed module $(M,G,\delta )$ is a covering pair of (G, N) if and only if there is a normal subgroup U of F with $M\cong Y/U$, $\ker \delta \cong (R\cap Y)/U$, and

$$\begin{aligned} \frac{R\cap Y}{[R,Y][F,R\cap Y]}=H_2(G,N)\times \frac{U}{[R,Y][F,R\cap Y]}. \end{aligned}$$

(4)

Proof

Let $(M,G,\delta )$ be a covering pair of (G, N). Then $(\ker \delta ,1,1)\rightarrowtail (M,G,\delta )\twoheadrightarrow (N,G,i)$ is a stem extension of the crossed module (N, G, i) and so, according to Mohammadzadeh et al. [19, Lemma 3.3], we find an epimorphism $\beta :Y/([R,Y][F,R\cap Y])\longrightarrow M$ such that $\beta (H_2(G,N))=\ker \delta$. Set $\ker \beta =U/([R,Y][F,R\cap Y])$ for some normal subgroup U of Y, then $M\cong Y/U$ and $\ker \delta \cong (R\cap Y)/U$. Also, the finiteness of $H_2(G,N)$ ensures that $H_2(G,N)\cap \ker \beta =1$. As the kernel of the restriction of $\beta _1$ to $(R\cap Y)/([R,Y][F,R\cap Y])$ is $\ker \beta$ and the image of this restriction is $\ker \delta$, we conclude that U satisfies the condition (4).

Conversely, let U be a normal subgroup of F satisfying (4), and $\bar{\mu }:Y/([R,Y][F,R\cap Y])\longrightarrow F/[F,R]$ be the crossed module induced by $\mu$. In view of [15, Theorems 2.1.4(i) and 2.4.6(iv)], there is a normal subgroup S of F such that $R/[F,R]=H_2(G)\times (S/[F,R])$. One easily sees that $\bar{\mu }(U/([R,Y][F,R\cap Y]))\subseteq S/[F,R]$. Using these results and Theorem 3.1(i), we obtain

$$\begin{aligned} \left( \frac{R\cap Y}{[R,Y][F,R\cap Y]},\frac{R}{[F,R]},\bar{\mu }\right) = H_2(N,G,i)\times \left( \frac{U}{[R,Y][F,R\cap Y]},\frac{S}{[F,R]},\bar{\mu }\right) . \end{aligned}$$

It, therefore, follows from [19, Proposition 3.1(i)] that $(Y/U,F/S,\bar{\mu })$ is a covering crossed module of (N, G, i) and so, the composite homomorphism $Y/U\longrightarrow Y/(R\cap Y){\mathop {\longrightarrow }\limits ^{\cong }}N{\mathop {\longrightarrow }\limits ^{i}}G$ is a covering pair of the pair (G, N), thanks to Lemma 4.1. The proof is complete. $\square$

The following corollaries are generalizations of the works of Jones and Wiegold [14], and Yamazaki [25].

Corollary 4.4

Let $(M_i,G,\delta _i)$ $i=1,2$, be two covering pairs of a pair (G, N) of finite groups. Then

(i) $M_1$ and $M_2$ are isoclinic.

(ii) $Z(M_1)/\ker \delta _1\cong Z(M_2)/\ker \delta _2$.

Proof

Suppose $(R\cap Y,R,\mu )\rightarrowtail (Y,F,\mu )\twoheadrightarrow (N,G,i)$ is a projective presentation of the inclusion crossed module (N, G, i). From Theorem 4.3 and its proof, we have an epimorphism $\beta :Y/([R,Y][F,R\cap Y])\longrightarrow M_1$ such that $(R\cap Y)/([R,Y][F,R\cap Y])=H_2(G,N)\times \ker \beta$. But this implies that

$$\begin{aligned} \left( \frac{Y}{[R,Y][F,R\cap Y]}\right) '\cap \ker \beta \subseteq \frac{[F,Y]}{[R,Y][F,R\cap Y]} \cap \ker \beta =1. \end{aligned}$$

So, by Hall [12, Page 134], $M_1$ and similarly, $M_2$ are isoclinic to $Y/([R,Y][F,R\cap Y])$, which proves (i).

To prove (ii), we only need to show that the factor $Z(M_1)/\ker \delta _1$ is determined uniquely by the free presentation $N\cong Y/(R\cap Y)$. Put

$$\begin{aligned} \ker \beta =\frac{U}{[R,Y][F,R\cap Y]}~~~~\mathrm{and}~~~~Z\left( \frac{Y}{[R,Y][F,R\cap Y]}\right) =\frac{W}{[R,Y][F,R\cap Y]} \end{aligned}$$

for some normal subgroups U and W of Y. Then $[Y,W]\subseteq U$ and so $W/U\subseteq Z(Y/U)$. On the other hand, if $xU\in Z(Y/U)$ then $[x,y]\in U\cap [Y,F]=[R,Y][F,R\cap Y]$ for all $y\in Y$. Hence

$$\begin{aligned} x[R,Y][F,R\cap Y]\in Z\left( \frac{Y}{[R,Y][F,R\cap Y]}\right) =\frac{W}{[R,Y][F,R\cap Y]}, \end{aligned}$$

which implies that $Z(Y/U)=W/U$. It, therefore, follows that $Z(M_1)/\ker \delta _1\cong W/(R\cap Y)$, as required. $\square$

Corollary 4.5

Any relative stem extension of a pair of finite groups is a homomorphic image of one of its covering pairs.

Proof

Let $(M,G,\delta )$ be an arbitrary relative stem extension of a pair (G, N) of finite groups, and let $(R\cap Y,R,\mu )\rightarrowtail (Y,F,\mu ){\mathop {\twoheadrightarrow }\limits ^{(\pi _1,\pi _2)}}(N,G,i)$ be a projective presentation of the inclusion crossed module (N, G, i). Due to Mohammadzadeh et al. [19, Lemma 3.3], there is the following commutative diagram with exact rows and surjective columns.

where $\beta |$ is the restriction of $\beta$. Put $\ker \beta |=\ker \beta =T/([R,Y][F,R\cap Y])$ for some normal subgroup T of Y. It can be seen in the proof of Theorem 4.3 that

$$\begin{aligned} \frac{R\cap Y}{T}\cong \ker \delta \cong \frac{R\cap [F,Y]}{T\cap R\cap [F,Y]} \cong \frac{T(R\cap [F,Y])}{T}, \end{aligned}$$

which, because of the finiteness of $\ker \delta$, implies that $T(R\cap [F,Y])=R\cap Y$. Also, the triviality of the second homology of F/Y yields that $Y\cap F'=[F,Y]$. So, we have

$$\begin{aligned} \frac{T}{T\cap [F,Y]}=\frac{T}{T\cap R\cap [F,Y]}\cong \frac{T(R\cap [F,Y])}{R\cap [F,Y]} =\frac{R\cap Y}{R\cap Y\cap F'}\cong \frac{(R\cap Y)F'}{F'}\le \frac{F}{F'}. \end{aligned}$$

Thus the following exact sequence of abelian groups splits

$$\begin{aligned} \frac{T\cap [F,Y]}{[R,Y][F,R\cap Y]}\rightarrowtail \frac{T}{[R,Y][F,R\cap Y]} \twoheadrightarrow \frac{T}{T\cap [F,Y]}, \end{aligned}$$

and hence

$$\begin{aligned} \frac{T}{[R,Y][F,R\cap Y]}=\frac{T\cap [F,Y]}{[R,Y][F,R\cap Y]}\times \frac{U}{[R,Y][F,R\cap Y]}, \end{aligned}$$

where $U/([R,Y][F,R\cap Y])\cong T/(T\cap [F,Y])$. Using these results, we have

$$\begin{aligned} U(R\cap [F,Y])=R\cap Y~~~~\mathrm{and}~~~~U\cap (R\cap [F,Y])=[R,Y][F,R\cap Y], \end{aligned}$$

which show that U satisfies (4). Accordingly, owing to Theorem 4.3, the map $\delta :Y/U\longrightarrow G$ induced by $\beta$ is a covering pair of (G, N) and, moreover, M is a homomorphic image of Y/U, which completes the proof. $\square$

Given a pair (G, N) of groups, it is shown in Lemma 4.1 that any stem cover of the inclusion crossed module (N, G, i) gives a covering pair for (G, N). Now, we prove the converse in the case where G is finite.

Proposition 4.6

Let (G, N) be a pair of finite groups. Then any covering pair of (G, N) can be lifted to a stem cover of the inclusion crossed module (N, G, i).

Proof

Let $(M,G,\delta )$ be a covering pair of (G, N). By Mohammadzadeh et al. [19, Theorem 3.4], there is a stem cover

$$\begin{aligned} (\hat{A},\hat{B},\hat{\delta })\rightarrowtail (\hat{M},\hat{G},\hat{\delta }) {\mathop {\twoheadrightarrow }\limits ^{(\psi _1,\psi _2)}}(N,G,i), \end{aligned}$$

and a surjective morphism $(\beta _1,\beta _2):(\hat{M},\hat{G},\hat{\delta })\longrightarrow (M,G,\delta )$ such that the diagram

is commutative. Invoking Lemma 4.1, $(\hat{M},G,i\psi _1)$ is a covering pair of (G, N). Since the groups M and $\hat{M}$ have the same order, it follows that $\beta _1$ is an isomorphism. Now, the composite homomorphism $\hat{\delta }\beta _1^{-1}:M\longrightarrow \hat{G}$, together with the action of $\hat{G}$ on M defined by $^{\hat{g}}m=~^{\beta _2(\hat{g})}m$ is a crossed module. Because, if $m,m_1,m_2\in M$, $\hat{g}\in \hat{G}$ and $\beta _1(\hat{m})=m$ for some $\hat{m}\in \hat{M}$, then using the above diagram we have

$$\begin{aligned}&\hat{\delta }\beta _1^{-1}\big (^{\hat{g}}m\big )= \hat{\delta }\beta _1^{-1}\big (^{\beta _2(\hat{g})}\beta _1(\hat{m})\big )= \hat{\delta }\beta _1^{-1}\big (\beta _1(^{\hat{g}}\hat{m})\big )= \hat{\delta }\big (^{\hat{g}}\hat{m}\big )=~^{\hat{g}}\hat{\delta }\beta _1^{-1}(m),\\&\quad ~^{\hat{\delta }\beta _1^{-1}(m_1)}m_2= ~^{\beta _2\hat{\delta }\beta _1^{-1}(m_1)}m_2= ~^{\delta (m_1)}m_2=~^{m_1}m_2. \end{aligned}$$

It is routine to verify that $\ker \delta \subseteq [\hat{G},M]\cap M^{\hat{G}}$, $\hat{B}=\ker \beta _2\subseteq Z(\hat{G})\cap st_{\hat{G}}(M)\cap \hat{G}'$ and $(\beta _1,id):(\hat{A},\hat{B},\hat{\delta })\longrightarrow (\ker \delta ,\ker \beta _2,\hat{\delta }\beta _1^{-1})$ is an isomorphism. We therefore conclude that the extension

$$\begin{aligned} \big (\ker \delta ,\ker \beta _2,\hat{\delta }\beta _1^{-1}\big ) \rightarrowtail \big (M,\hat{G},\hat{\delta }\beta _1^{-1}\big ) {\mathop {\twoheadrightarrow }\limits ^{(\delta ,\beta _2)}}(N,G,i) \end{aligned}$$

is a stem cover of (N, G, i). Moreover, $(M,G,\delta )$ and $(M,G,i\delta )$ are isomorphic. $\square$

We end this article with the following interesting corollary.

Corollary 4.7

Let G be a finite group with a normal subgroup N such that the inclusion crossed module (N, G, i) is perfect. Then

(i) All covering pairs of the pair (G, N) are isomorphic.

(ii) A relative central extension of (G, N) is universal if and only if it is a covering pair.

(iii) Every universal central extension of (G, N) can be lifted to a universal central extension of (N, G, i).

Proof

(i) Combine the above proposition with [24, Corollary 3.8].

(ii) In the proof of Corollary 3.8, it was shown that the perfect crossed module (N, G, i) admits a universal central extension, which is, up to isomorphism, its unique stem cover. The result now follows from these facts, Lemma 4.1, part (i), and the uniqueness of the universal relative central extension.

(iii) This follows from part (ii) and Proposition 4.6. $\square$