.I have 2 questions on cosets.

Let be a group with subgroup .

Let .

If there exists such that , show that .

I have tried this problem for a long time, but I still can't get a convinced proof.

Please give me some hint.

I assume your "a" must in fact be "g", so , so if then

, but

this last rightmost expression is in N (why? Hint: normality), so we get a contradiction.

Another question is if , is it true that ?

Not at all: and that's all.

For example, take

Tonio