Past exam of the groups and representations course of the University of Oxford / 2020 / 3 / a / Solution

The componentwise product is closed in $G_1\times G_2$. Associativity follows componentwise from associativity in each group:The identity is $(e_1,e_2)$, and $(g_1,g_2{)}^{-1}=(g_1^{-1},g_2^{-1})$. These verify all the group axioms, so this is the direct product of groups.

Write the paper's Kronecker product $\times$ as $\otimes$. Its mixed-product identity and the representation laws of $R_1,R_2$ giveAlso $R(e_1,e_2)=I_{n_1}\otimes I_{n_2}=I_{n_1{n}_2}$, andThus the images are invertible and this is a group representation of dimension $n_1{n}_2$, called the external tensor product of group representations. No irreducibility assumption is needed for this construction.