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.

Many Thanks.