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<h2>About HAP: The nonabelian tensor product of crossed modules</h2>

<p>
The right-hand side of the formula
$$\pi_3(SK(G,1)) \cong \ker (G\otimes G \rightarrow G, g\otimes g' \mapsto gg'g^{-1}g'^{-1})$$
involves a special case of a crossed module
$$M\otimes N \rightarrow G, m\otimes n \mapsto (\partial m)(\partial'n)(\partial m)^{-1}(\partial' n)^{-1}$$
arising from two crossed modules $\partial\colon M\rightarrow G$ and
$\partial'\colon N\rightarrow G$ with common target group $G$.
Here $\otimes$ denotes the nonabelian tensor product introduced by Ronnie Brown and Jean-Louis Loday.
</p>

<p>
The following two commands construct the crossed modules $M\hookrightarrow G$
and $N\hookrightarrow G$ arising from two normal subgroups $M$, $N$ of the sylow 2-subgroup $G=Syl_2(M_{24})$ of the Mathieu group $M_{24}$, and then compute the homotopy groups $\pi_i(M\otimes N \rightarrow G)$ for
$i=1,2$.
</p>

<div><code>
gap> G:=SylowSubgroup(MathieuGroup(24),2);;<br>
gap> L:=NormalSubgroups(G);;<br>
gap> M:=CrossedModuleByNormalSubgroup(G,L[100]);;<br>
gap> N:=CrossedModuleByNormalSubgroup(G,L[101]);;<br>
gap> T:=NonabelianTensorProduct(M,N);;<br>
gap> StructureDescription(HomotopyGroup(T,1));<br>
"((C2 x ((C2 x C2 x C2 x C2) : C2)) : C2) : C2"<br>
gap> StructureDescription(HomotopyGroup(T,2));<br>
"C2 x C2 x C2 x C2 x C2 x C2 x C2 x C2"<br>
</code></div>

<p>
The above construction can be iterated. In particular, one can construct the triple tensor product $(G\otimes G)\otimes G \longrightarrow G$ for any group $G$. The following commands compute
$$\exp((G\otimes G)\otimes G) = 4 \, ,$$
$$|(G\otimes G)\otimes G| = 1073741824 \, ,$$
for $G$ the 23rd group of order 32 in GAP's library of small groups.
</p>

<div><code>
gap> G:=SmallGroup(32,23);;<br>
gap> K:=CrossedModuleByNormalSubgroup(G,G);;<br>
gap> T2:=NonabelianTensorProduct(K,K);;<br>
gap> T3:=NonabelianTensorProduct(T2,K);;<br>
gap> T:=Source(T3!.map);;<br>
gap> Exponent(T);<br>
4<br>
gap> Order(T);<br>
1073741824<br>
</code></div>

<p>
Corresponding to the nonabelian tensor product, there is  a nonabelian exterior product
$$M\wedge N \rightarrow G, m\wedge n \mapsto (\partial m)(\partial'n)(\partial m)^{-1}(\partial' n)^{-1}$$
arising from two crossed modules $\partial\colon M\rightarrow G$ and
$\partial'\colon N\rightarrow G$ with common target group $G$.
The exterior product is obtained from the tensor product by adding the relation
$$m \otimes n = 1\ \ {\rm whenever}\ \partial m=\partial'n\,.$$
</p>

<p>
This exterior product can be  iterated. In particular, one can construct the triple exterior product $(G\wedge G)\wedge G \longrightarrow G$ for any group $G$. The following commands compute
$$\exp((G\wedge G)\wedge G) = 4 \, ,$$
$$|(G\wedge G)\wedge G| = 1024 \, ,$$
for $G$ the 23rd group of order 32 in GAP's library of small groups.
</p>

<div><code>
gap> G:=SmallGroup(32,23);;<br>
gap> K:=CrossedModuleByNormalSubgroup(G,G);;<br>
gap> E2:=NonabelianExteriorProduct(K,K);;<br>
gap> E3:=NonabelianExteriorProduct(E2,K);;<br>
gap> e:=Source(E3!.map);;<br>
gap> Exponent(e);<br>
4<br>
gap> Order(e);<br>
1024<br>
</code></div>

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