While dealing with a question on the forum, I came up with the following problem:

If is a subgroup of a group , and , does imply ? In other words, is it possible to have ?

The answer (to the last question) is "no" if is finite. Or if is a normal subgroup. But what else?

I think this is either easy or false, but I can find neither a proof nor a counterexample...

I hope some people can help,

Thanks in advance,

Laurent.