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$20=2^2\times5 $

My initial reaction was to write direct products between an abelian and non abelian groups — for example, $Z_5\times D_2$. However, I realized they are both abelian so I am a little stuck. I can't think of any subgroup that is not abelian. Could it be a semidirect product or something? If so, could you give me a hint? My knowledge of semidirect products is sketchy to say the least.


marked as duplicate by Derek Holt group-theory Nov 11 '15 at 8:45

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  • $\begingroup$ $D_4$ is not abelian $\endgroup$ – Kushal Bhuyan Nov 11 '15 at 5:21
  • $\begingroup$ Pardon me I meant to write $D_2$ because $|D_2|$=4 $\endgroup$ – user139985 Nov 11 '15 at 5:31
  • $\begingroup$ then edit your question please. $\endgroup$ – Kushal Bhuyan Nov 11 '15 at 5:31
  • $\begingroup$ One obvious one is the dihedral group of order 20, which is the semidirect product of $Z_5$ with $Z_2\times Z_2$. There ought to be a couple of semidirect products of $Z_5$ with $Z_4$ depending on whether the (nontrivial) map from $Z_4$ to Aut $Z_5$ is an isomorphism or maps to a subgroup of order 2. $\endgroup$ – John Brevik Nov 11 '15 at 5:35
  • $\begingroup$ To classify all nonabelian groups of order $20$, you should use the Sylow Theorems. What can you say about the number of Sylow $2$- and $5$-subgroups? $\endgroup$ – André 3000 Nov 11 '15 at 5:45

take $D_5\times Z_2$. $D_5$ is non-abelian.

  • $\begingroup$ For some reason I was thinking you had to make subgroups only of 4 and 5 or 2,2 and 5, but I think you're right. Would this be the only one? $\endgroup$ – user139985 Nov 11 '15 at 5:39
  • $\begingroup$ that I am not sure but ...every group of order $2,3,4,5$ is abelian $\endgroup$ – Black-horse Nov 11 '15 at 5:42

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