having some trouble with this question.
For one, I don't understand how the direct sum can operate across groups, so im assuming were direct summing the regular module of these groups
Second, I don't have much clue what would be generated by this direct sum, be it a module or otherwise, and I haven't got much clue how to figure it out.
Third, I'm not entirely sure what it means to simultaneously diagonalize a module.
Thanks for any help you can provide!
I should add this is a repost of a question I posed 2 days ago which went unseen, hoping someone who can/will help will see it this time!