Let $N$ be finite abelian $p$-group (is not cyclic). Is there any extension (not central extension) of $N$ by $A_{5}$?
Extension of a p-group abelian by simple group
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group-theory
p-groups
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0An idea: since you don't want a central extension, direct products are out. The next natural candidate is a non-trivial semidirect product. Since $\,Aut(A_5)\cong S_5\,$ we have quite some possibilities here for some non-cyclic abelian p-groups. My first thought goes to the Klein viergrup. – 2012-10-22
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0You also can consider $N$ is non-abelian. – 2012-10-22
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1When you say an "extension of $N$ by $A_5$", do you mean: (i) a group with $G$ with a normal subgroup $K$ with $K \cong N$ and $G/K \cong A_5$; or (ii) a group with a normal subgroup $K$ with $K \cong A_5$ and $G/K \cong N$. Unfortunately, both meanings arise in the literature, with roughly equal frequency! I am guessing you mean (i), since otherwise there would be possibility of it being a central extension. Also is $N$ intended to be a specific abelian $p$-group or an arbitrary abelian $p$-group. In the first case, the answer is that it depends on $N$. – 2012-10-22