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I have been doing some studying and I was wondering if there was any literature regarding the conjugacy classes size in a finite nilpotent group in determining if a group is a p-group or not. It seems to be that there is not; I was wondering if anyone knew of any literature (I can not seem to locate anything through online resources and my library). If there is no literature I am curious as to why because it seems that this would be a studied question.

EDIT: To be more precise. The question is Let G be finite nilpotent group. Given the largest class size can one determine if G is of prime power order.

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    $\begingroup$ I don't think so, if I understand the question correctly. Suppose that $G = P\times H$, with $P$ (for instance) a non-abelian $3$-group, and $H$ an abelian $2$-group. Then $G$ is nilpotent, not a $p$-group, for any $p$, and the largest class size in $G$ is the same as the largest class size in $P$. $\endgroup$
    – James
    Nov 21 '17 at 14:12
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I don't know what you mean by using the sizes of conjugacy classes to determine if a group is a $p$-group, but it's a standard theorem that finite nilpotent groups are precisely those groups which are a product of their Sylow subgroups, i.e., a product of $p$-groups.

Moreover, the conjugacy classes of $A \times B$ for $A, B$ groups are precisely the pairs of conjugacy classes of $A$ and $B$. So for a nilpotent group, the sizes of the conjugacy classes that you get will be the product of the sizes for each $p$-group.

Does that help answer your question?

Edit: James has pointed out that you cannot characterize the $p$-groups using only the set of cardinalities of conjugacy classes. I'll just add that if you record the multiset of cardinalities of conjugacy classes, then obviously you can add them up to get the order of the group.

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  • $\begingroup$ My question is incredibly unclear, I am sorry for that. I will edit my question to make it more precise and try to be more clear. $\endgroup$
    – user499340
    Nov 21 '17 at 13:34

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