Give examples of two non-isomorphic groups of order $n^2$

Question

For each integer, $n > 7$, give examples of two non-isomorphic groups of order $n^2$.

I wonder why the questions asks for $n > 7$. That seems somewhat random. — Tobias Kildetoft, Dec 04 '13 at 19:10

score 6 · Accepted Answer · answered Dec 04 '13 at 18:57

6

Hint: For every $m$ there is a group of order $m$ (the cyclic one), so you have one group of order $n^2$ already. Now you just need another one, and a reason why it is not isomorphic to the one you already have. Think of $\mathbb Z_n\times \mathbb Z_n$.

answered Dec 04 '13 at 18:57

Ittay Weiss

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score 4 · Answer 2 · answered Dec 04 '13 at 18:57

4

How about $\mathbb{Z}/n^{2}\mathbb{Z}$ and $\mathbb{Z}/n\mathbb{Z}\oplus \mathbb{Z}/n\mathbb{Z}$. One is cyclic and the other not.

answered Dec 04 '13 at 18:57

TheNumber23

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score 3 · Answer 3 · answered Dec 04 '13 at 18:59

3

Hint: Think of $\mathbb{Z}_n$ and direct product of this.

answered Dec 04 '13 at 18:59

LASV

3,108

Give examples of two non-isomorphic groups of order $n^2$

3 Answers3