Let $G$ be a finite group, generated by two elements. If $H\le G$ is a subgroup, must $H$ also be generated by 2 elements?
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2Excluding the trivial subgroup? – Sean Haight Nov 22 '16 at 01:37
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1Wouldn't any group then be generated by infinitely many (non-distinct) elements. – Sean Haight Nov 22 '16 at 01:43
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The symmetric groups $S_n$ are generated by two elements, namely the full cycle $(1\;2\;\ldots\;n)$ and the transposition $(1\;2)$. Every finite group is a subgroup of a symmetric group by Cayley's theorem. So, if the statement were true, every finite group would be generated by 2 elements, which is clearly false.
marlu
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The smallest group which is not generated by two elements is $C_2^3$, with order $8$, and the smallest symmetric group in which it embeds is $S_6$. But one can embed it into something smaller; in fact the semidirect product $C_2^3 \rtimes C_3$, where $C_3$ acts cyclically, is generated by two elements, and $C_2^3$ embeds into it. – Qiaochu Yuan Nov 22 '16 at 02:16