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The non abelian exterior square of a group is defined Non abelian exterior square of a group.. The presentation for alternating group on four symbols $\{1,2,3,4\}$ is $A_4=\langle (12)(34), (123)\rangle$. We have a well known onto homomorphism from $A_4\wedge A_4$ to $[A_4,A_4]$, by $g\wedge h \mapsto [g,h]$. In this case $(12)(34)\wedge (123)\mapsto [(12)(34), (123)]=(14)(23)$. Now I want to check that whether $(12)(34)\wedge (123)\mapsto (13)(24)$ becomes a homomorphism or not?

I am trying to do it by calculation but its very cumbersome. Is there any easy technique to check it. In fact I want to count number of homorphisms from $A_4\wedge A_4$ to $[A_4,A_4]$.

Shaun
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MANI
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    I saw a proof in the arXiv paper arXiv:1912.06080v1. There $Q_{2n}$ has order $8n$, so they denote by $Q_2$ what usually is $Q_8$. But I agree (see your link), that one should not write $A_4\ast A_4$ there. – Dietrich Burde Sep 19 '22 at 14:21
  • @DietrichBurde its just symbols, I have seen in some books they use $Q_8$, but I have edited that now and arXiv paper arXiv:1912.06080v1 is mine paper. – MANI Sep 19 '22 at 14:30
  • @DietrichBurde It will be so kind of you, if you can give some suggestions to me. – MANI Sep 20 '22 at 10:53

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