Number of conjugacy classes in a direct product is the product of the number of conjugacy classes in each factor

From Groupprops
Jump to: navigation, search
This article gives an expression for the value of the arithmetic function number of conjugacy classes of an external direct product in terms of the values for the direct factors. It says that the value for the direct product is the product of the values for the direct factors.
View facts about number of conjugacy classes: (facts closely related to number of conjugacy classes, all facts related to number of conjugacy classes)
View facts about external direct product: (facts closely related to external direct product, all facts related to external direct product)
View facts about product: (facts closely related to product, all facts related to product)

Statement

For two groups

Suppose G_1 and G_2 are groups. Suppose the number of conjugacy classes in G_1 is a_1 and the number of conjugacy classes in G_2 is a_2.

Then, the number of conjugacy classes in the external direct product G_1 \times G_2 is the product a_1a_2.

Note that this says that when either G_1 or G_2 has infinitely many conjugacy classes, so does G_1 \times G_2, and the product statement is interpreted in the sense of infinite cardinals. If both G_1 and G_2 have finitely many conjugacy classes, then so does G_1 \times G_2.

For multiple groups

Suppose G_1,G_2,\dots,G_n are groups. Suppose the number of conjugacy classes in G_1 is a_1, in G_2 is a_2, and so on.

Then, the number of conjugacy classes in the external direct product G_1 \times G_2 \times \dots \times G_n is the product a_1a_2 \dots a_n.

If all the a_i are finite, then so is the number of conjugacy classes in the direct product, and the multiplication is the usual multiplication of natural numbers. If one or more of the a_i is infinite, then the number of conjugacy classes in the direct product is also infinite, and the product is interpreted in terms of infinite cardinals.

Note that the statement also holds for an internal direct product, because of equivalence of internal and external direct product.

Related facts