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If we have two groups $G,G^{'} $ of order $n$ and both have the same number of elements of a given order then are the two groups isomorphic?

I meant that if $o(G)=o(G^{'})=n$ and if $G$ has $p$ elements of order $m$ then $G^{'}$ has also $p$ elements of order $m$ and this holds for each $m\in \Bbb N$ , then are $G,G^{'} $ isomorphic?

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  • You mean $G$ has $p$ elements of order $m$ if and only if $G'$ has $p$ elements of order $m$ for every $m$? – Couchy Jan 14 '17 at 18:53
  • They are finite groups, aren't they? – ajotatxe Jan 14 '17 at 18:54
  • Yes they are@ajotatxe – Learnmore Jan 14 '17 at 18:55
  • Seems like induction on order of the groups and quotients sholud do the work... (Only guessing). – ajotatxe Jan 14 '17 at 18:56
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    Strongly related: https://math.stackexchange.com/questions/729611, https://math.stackexchange.com/questions/1296833 – Watson Jan 14 '17 at 18:56
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    In the Abelian case, yes, in the non-Abelian case, no. See http://math.stackexchange.com/questions/693163/groups-with-same-number-of-elements-of-each-order and http://math.stackexchange.com/questions/72944/two-finite-abelian-groups-with-the-same-number-of-elements-of-any-order-are-isom – sTertooy Jan 14 '17 at 18:59

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