The Irreps of the Product of Finitely Many Finite Groups (Pt. II)
Point of post: This is a continuation of this post.
Corollary: Let be finitely many finite groups then,
From the above it’s clear that we may index the elements of as and so we have the characters of may be written as and we have that
The last thing we note which can often come in hand for practical computation (since, for instance it can be done on math world) is the following:
Theorem: Let be a finite group with irreducible characters and conjugacy classes . Moreover, let a finite group with irreducible characters and conjugacy classes . If is character table of thought of as a matrix with entry and is the character table of thought of as the matrix with entry . Then, if is the character table of where the rows are arranged left to right and the columns arranged up to down . Then, where is the Kronecker product.
Proof: This follows immediately from the definition
1. Simon, Barry. Representations of Finite and Compact Groups. Providence, RI: American Math. Soc., 1996. Print.