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Let $G=\{a_1,a_2,\ldots,a_n\}$ be an abelian group and let $m_1\leq m_2\leq \ldots \leq m_n$ be the list of orders of all the elements of $G$.

E.g., for the Klein 4-group $V$ we get the list $1,2,2,2,$ and for the cyclic group $C_4$ we get the list $1,2,4,4$.

Does the list of orders determine $G$? If not, is it true for $p$-groups?

Thanks!

Shaun
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boaz
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    here is a link to a paper claiming to have proven that (for abelian groups) the order sequence determines the group. (I have not checked the proof). The claim is not true for non-abelian groups, there are counterexamples for groups of order $16$, for example. (see, e.g., this). – lulu Feb 17 '19 at 17:41
  • Thanks @lulu for this information! – boaz Feb 17 '19 at 17:48

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