I am a student who has just started abstract algebra.
It's a simple question for you.
To let $ X_1 le Y $ and $ X_2 $ is a (normal or ideal) of $ Y $
Loud $ 3rd $ Isomorphism Theorem, We can easily conclude that $ X_1 cap X_2 $ is an ideal of $ X_1 $,
But the question is
$ (1) $ does $ X_1 cap X_2 $ is an (ideal or normal subgroup) of $ X_2 $?
$ (2) $ does $ X_1 cap X_2 $ is an (ideal or normal subgroup) of $ Y $?
Whenever you try to prove (1) and (2), they look like true.
But I have no confidence that both are true.
Any help would be appreciated.