It is of interest to me to find if there is a way to reconstruct the group knowing the (multi)-set of its conjugacy class sizes or, in other words, given the indices of all centralizers is there a unique group satisfying them (up to isomorphism)?

A piece of motivation was the similar statement for graphs involving degree sequences. It is known that non-isomorphic graphs with equal degree sequences do exist in that case.

The answer to my question in general is no as well since I've noticed that both dihedral group $D_8$ and Quaternion group $Q_8$ have the same multiset $\{ 1, 1, 2, 2, 2 \}$ although they are obviously not isomorphic.

However are there any well-known results in this area? E.g. for groups of particular order or with some kind of restriction applied?

  • $\begingroup$ Do you mean like groups of prime order? $\endgroup$ – Somos Apr 22 '18 at 2:26
  • $\begingroup$ @Somos that would be a nice one! Another idea is to find a property possessed by only one of 2 groups mentioned previously. Restricting groups to those having the property could yield further results. $\endgroup$ – Kater Murr Apr 26 '18 at 22:30
  • $\begingroup$ I meant to imply that any abelian group has all conjugacy classes of size 1. Thus, any group with a prime number of conjugacy classes all of size 1 is the unique cyclic group of that order. $\endgroup$ – Somos Apr 26 '18 at 22:58

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