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To a finite $p$-group, we can associate two vectors $(v_0,v_1,\dotsc)$:

  1. The class vector - $v_i$ is the number of conjugacy classes of order $p^i$.

  2. The character vector - $v_i$ is the number of complex irreducible representations of dimension $i$ up to isomorphism.

Question: Do these two invariants of the group determine each other? In other words, if two groups have the same class vector, do they have the same character vector and vice versa?

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  • $\begingroup$ @ToddLeason Should have tried that before asking. $\endgroup$
    – Amritanshu Prasad
    Commented Dec 20, 2018 at 3:58

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The answer is no. This observation is cataloged at the Group Properties Wiki on pages 1 and 2.

It is possible to have two finite groups $G_1$ and $G_2$ such that the conjugacy class size statistics of $G_1$ are the same as those of $G_2$ (i.e., the two groups have the same number of conjugacy classes of each size) but the Degrees of irreducible representations over $\mathbb{C}$ for $G_1$ are not the same as those of $G_2$...

It is possible to have two finite groups $G_1$ and $G_2$ such that the multiset of Degrees of irreducible representations (over $\mathbb{C}$) of $G_1$ is the same as the multiset of degrees of irreducible representations of $G_2$ (i.e., $G_1$ and $G_2$ have the same number of irreducible representations of each degree) but the conjugacy class size statistics of $G_1$ and $G_2$ are not the same...

Further, the smallest example of the first statement is of size $128$, while for the other statement, examples of groups of size $64$ exist so that these statements are true for $p$-groups as well.

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    $\begingroup$ In the GAP small groups library, the groups with ID (128,1758) and (128,2022) have the same class vector: $(2,14,48,64)$. However, their character vectors are $(16,12,0,1)$ and $(16,8,5,0)$, respectively. $\endgroup$
    – Amritanshu Prasad
    Commented Dec 21, 2018 at 4:27
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    $\begingroup$ In the GAP small groups library, the groups with ID (64,4) and (64,257) have the same character vectors (16,4,2), but their class vectors are (4,6,12) and (2,9,11) respectively. $\endgroup$
    – Amritanshu Prasad
    Commented Dec 26, 2018 at 11:48

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