Intersection of two normal subgroups of a group

Question

Let G be a group, and let A,B be normal subgroups of G. If $a \in A$ and $b \in B$, does this mean that $ab \in A \cap B$?

Answer 1

If $G$ has more than one element, then this is never true. $A = G$ and $B = \{1_G\}$ are normal subgroups of $G$. Let $a \in A$ be any element which is not the identity, and let $b = 1_G \in B$. Then $ab = a$ does not lie in $A \cap B = \{1_G\}$.

Maybe you are thinking of the analogue of normal subgroups in ring theory, that is ideals. If $I, J$ are ideals of a (commutative) ring $R$, and $a \in I, b \in J$, then certainly $ab \in I \cap J$.