Classify the non-abelian subgroups of order 20.

Question

$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.

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. — John Brevik, Nov 11 '15 at 05:35
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? — Viktor Vaughn, Nov 11 '15 at 05:45

score 0 · Answer 1 · answered Nov 11 '15 at 05:34

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

answered Nov 11 '15 at 05:34

Black-horse

1,441

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? – user139985 Nov 11 '15 at 05:39
that I am not sure but ...every group of order $2,3,4,5$ is abelian – Black-horse Nov 11 '15 at 05:42

Classify the non-abelian subgroups of order 20.

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