Degree of irreducible representation need not divide order of derived subgroup

From Groupprops
Jump to: navigation, search

Statement

It is possible to have a finite group G and an irreducible linear representation \varphi of G over a splitting field in characteristic zero such that the degree of \varphi does not divide the order of the Derived subgroup (?) of G.

This is a numerical non-constraint on the Degrees of irreducible representations (?).

Related facts

For more facts, see degrees of irreducible representations.

Similar facts

Opposite facts

Related facts about conjugacy class sizes

These facts are related to the conjugacy class size statistics of a finite group.

Proof

Example of symmetric group of degree three

Further information: symmetric group:S3, subgroup structure of symmetric group:S3, linear representation theory of symmetric group:S3

We consider the group symmetric group:S3. The standard representation of symmetric group:S3 is a faithful irreducible representation of degree two, and the derived subgroup, A3 in S3, has order three.

Example of extraspecial group

Let p be a prime number. Consider an extraspecial group of order p^5 for any prime number p. The derived subgroup has order p. However, this group has a faithful irreducible representation of degree p^2.