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Give four groups of order 20 that are not isomorphic.

I know the integers under addition mod 20 is one group of order 20, but what would three other groups of order 20 that are not isomorphic to it?

hersch
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1 Answers1

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  • We can easily find two groups of order $20$ (even two abelian groups of order $20$) that are not isomorphic: $$\mathbb Z_{20} \not\cong \mathbb Z_{2}\times \mathbb Z_{10}$$

    We know this because $$\mathbb Z_{mn} \cong \mathbb Z_m \times \mathbb Z_n \iff \gcd(m, n) = 1$$

  • Also add the Dihedral group of order $20$: The group of symmetries of a regular decagon.

  • You might want to visit the Groupprops website, now or in the future. It comes in very handy for problems of this sort, but also as a handy reference for group theory (definitions, theorems, classification of groups, etc): Groupprops: groups of order 20.

There are exactly FIVE non-isomorphic groups of order $20$. The two not already mentioned are the dicyclic group $\mathrm{Dic}_{20},$ and the general affine group $\mathrm{GA}(1, 5)$.

amWhy
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    There's also the dihedral group of order $20,$ which is non-abelian, so different from the cyclic group of order $20$ and the direct product of cyclic groups you mention. The other two involve non-trivial semi-direct products. – Cameron Buie Nov 26 '13 at 17:31
  • Yes, @Cameron, as noted on the dihedral group. Thanks for the added reference to the semi-direct products. – amWhy Nov 26 '13 at 17:33
  • You are welcome. (+1) – Cameron Buie Nov 26 '13 at 17:45
  • @amWhy: Another UV on the way! +1 – Amzoti Nov 27 '13 at 03:53
  • hersch - Is there a reason you haven't accepted the answer? You can accept an answer even though it's on hold, or a duplicate, if it's been helpful! ;-) – amWhy Dec 03 '13 at 17:36