Itô's theorem

This article is about the result in representation theory. For the result in stochastic calculus, see Itô's lemma.

Itô's theorem is a result in the mathematical discipline of representation theory due to Noboru Itô. It generalizes the well-known result that the dimension of an irreducible representation of a group must divide the order of that group.

Statement

Given an irreducible representation V of a finite group G and a maximal normal abelian subgroup A ⊆ G, the dimension of V must divide [G : A].

References

• James, Gordon; Liebeck, Martin (1993). Representations and Characters of Groups. Cambridge University Press. p. 247. ISBN 0-521-44590-6.
• Weisstein, Eric. "Itô's Theorem". Wolfram Mathworld. Wolfram Research. Retrieved 6 November 2018.