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So i am trying to prove that a non-abelian group with 362 elements is isomorphic to the dihedral group 181 (D181)? So the dihedral group has 362 elements(181*2). Exactly the same as the non-abelian group. And the elements of D181 are all matrices. Dn is non-abelian for every n (except 1,2). How can i map the elements from the non-abelian group to the dihedral group? I can't figure it out. Any tips?

J. W. Tanner
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Simon Jachson
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    N.B.: $181$ is prime, so see here – J. W. Tanner May 09 '19 at 15:47
  • yes, and how does that help me? – Simon Jachson May 09 '19 at 17:14
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    It helps you by answering your question. The point is that it is easier and more useful to solve this problem for groups of order $2p$ for $p$ prime, rather than just to do it in the special case when $p=181$. – Derek Holt May 09 '19 at 18:04
  • For an explicit isomorphism, you can map any element of order $181$ in one group to any element of order $181$ in the other, and any element of order $2$ in one group to any element of order $2$ in the other. (It is not always as easy as that to find an isomorphism between two groups.) – Derek Holt May 09 '19 at 18:08

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Note that $362=2\times181$ and $181$ is prime.

As shown here, groups of order $2p$ with $p$ prime are either cyclic or dihedral,

so if it's non-Abelian, it must be dihedral.

J. W. Tanner
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