## Interaction of the left and right regular representations of a group

Let be the left regular representation of and let be the right regular representation. That is, for all , and .

- Prove that and commute for all . (Thus and .)
- Prove that if and only if is an element of order 1 or 2 and .
- Prove that if and only if and . Deduce that .

- Let . Now . Thus for all . In other words, and .
- Suppose . Then for all , . Letting we see that ; thus either or . Moreover, we have for all ; thus .
Suppose is either 1 or has order 2. Then . Then we have for all ; hence .

- Suppose . Then for all , . For this gives . Thus we have and for all ; hence .
Suppose and . Then for all , we have ; thus .

Now if , then for some . As we saw above, . Thus is contained in both and .

Suppose . Then for some . Now , hence . Similarly, .

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## Comments

In my book, is not in 3. Is it an error in the book?

This is from the errata.

I haven’t checked myself, but I imagine that the statement is false if we drop .