.I have 2 questions on cosets.
Let be a group with subgroup .
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
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