Conjugacy class sizes of direct product are pairwise products of conjugacy class sizes of direct factors

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Statement

For two groups

Suppose G_1 and G_2 are finite groups.Suppose c_{11},c_{12},\dots,c_{1r} are the sizes of the conjugacy classes of G_1 (with repetitions, i.e., if a particular conjugacy class size occurs for multiple conjugacy classes, it appears that many times on the list) and c_{21},c_{22},\dots,c_{2s} are the sizes of the conjugacy classes of G_2.

Then, the conjugacy class sizes for the external direct product are given by taking pairwise products between conjugacy class sizes of G_1 and of G_2:

c_{11}c_{21}, c_{11}c_{22}, \dots,c_{11}c_{2s},c_{12}c_{21},c_{12}c_{22},\dots,c_{12}c_{2s},c_{1r}c_{21},\dots,c_{1r}c_{2s}

In particular, this means that the conjugacy class size statistics of G_1 \times G_2 are completely determined by the conjugacy class size statistics of G_1 and G_2, without any further information.

Due to the equivalence of internal and external direct product, this result also applies to internal direct products.

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