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I'm trying to exhibit two non-isomorphic non-abelian groups of order $243 = 3^5$.

I know that there are subgroups of order $3^k$ for $k$ ranging from $0$ to $5$ inclusive, where the subgroup of order $3$ is cyclic, but unsure how to proceed: I was also taking a look at a similar problem but feel like I might be missing something.

Any hints are appreciated; I was just interested since it's a practice prelim question. Thanks!

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2 Answers 2

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Hint For any prime $p$ there are (exactly) two nonabelian groups of order $p^3$ up to isomorphism.

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Trying to guess what you would expected to know at a prelim (may be you're in a program with more ivy than mine had, but anyway). Undoubtedly you have done the exercise of showing that matrices over $\Bbb{F}_3$ with the shape $$ \left(\begin{array}{ccc}1&*&*\\0&1&*\\0&0&1\end{array}\right) $$ form a non-abelian group of order $3^3$ (the exercise often requiring you to check that all the non-identity matrices are of order $3$)

Why don't you try shapes $$ \left(\begin{array}{ccccc}1&0&0&*&*\\0&1&0&0&*\\0&0&1&0&*\\ 0&0&0&1&*\\0&0&0&0&1\end{array}\right) $$ and $$ \left(\begin{array}{cccc}1&0&*&*\\0&1&*&*\\0&0&1&*\\ 0&0&0&1\end{array}\right)? $$

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