Proving $K(G_1 \times G_2) \cong KG_1 \otimes_K KG_2$, as K-algebras

Asked Oct 03 '15 at 16:04

Active Oct 03 '15 at 16:04

Viewed 65 times

I've seen a post (here) that the following isomorphism of $K$-algebras holds: $$K[G_1\times G_2] \cong K[G_1] \otimes_K K[G_2].$$ In Curtis and Reiner's Representation Theory of Finite Groups and Associative Algebras, an exercise asks to prove that $$K(G_1 \times G_2) \cong K[G_1] \otimes_K K[G_2].$$

This is probably more a question on notation, but what is $K(G_1 \times G_2)$?

Any help would be appreciated, thanks.

edited Apr 13 '17 at 12:20

Community

asked Oct 03 '15 at 16:04

user130075

Do you know what the direct product of groups is? – Pedro Oct 03 '15 at 16:56
1

Presumably the same thing. Possibly just as vector spaces. You didn't say what exercise. – anon Oct 03 '15 at 17:52
@whacka Sorry about that, the exercise is 12.8 (Chapter 12 is on tensor products) – user130075 Oct 04 '15 at 18:14

Proving $K(G_1 \times G_2) \cong KG_1 \otimes_K KG_2$, as K-algebras

0 Answers0