How does one prove that every finite group is isomorphic to a subgroup of an alternating group?
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firstly by trying, what have you tried? – mdave16 Aug 08 '17 at 23:46
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1Hint: it suffices to show it for symmetric groups. – lulu Aug 08 '17 at 23:47
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1https://math.stackexchange.com/questions/27132/embedding-of-finite-groups – Tsemo Aristide Aug 08 '17 at 23:48
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Possible duplicate of Embedding of finite groups – Ethan Bolker Aug 09 '17 at 00:06
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By Cayley's theorem every group is isomorphic to a subgroup of $S_n $.
Then $S_n $ is isomorphic to a subgroup of $A_{n+2}$. To see this, map even permutations to themselves, and for odd permutations map to the even permutation gotten by multiplying by the transposition $(n+1 \ n+2) $. It is straightforward to check that this defines an embedding.
One can take the composition of the two embeddings: $G\hookrightarrow S_n\hookrightarrow A_{n+2}$, where $n=|G|$.