Counter examples for normal subgroups I'm looking for some counterexamples to some particular questions to do with normality in group theory.Any help would be appreciated. 
Suppose H is a subgroup of a group G. 


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*If K is a normal subgroup of G. Show by counterexample that H ∩ K is not necessarily a normal subgroup of G

*For an arbitrary subgroup J. Show by counterexample that H ∩ J is not necessarily a normal subgroup of H. 


I'm not very experienced with group theory and am not really sure what sort of groups I should be looking at here. Thanks for any help.
 A: Example 1:
Let $G={\rm GL}_2(F)$ (where $F$ is any field) and let $K={\rm SL}_2(F)$, The subgroup $K$ is normal in $G$ as it is the kernel of the determinant map. Now let $H={\rm B}_2(F)$ the Borel subgroup of upper triangular matrices. Then $H\cap K$ is not normal in $G$ (nor in $K$).
Example 2:
Now take $G={\rm GL}_3(F)$, $H={\rm GL}_2(F)$ (seen as the subgroup of $G$ having a block $2\times2$ in the top left corner, a $1$ in the lower right corner, and $0$s elsewhere) and let $J=B_3(F)$ the upper triangular $3\times3$ matrices in $G$.
A: Any counterexample for your first statement is also a counterexample for your second statement (any normal subgroup is also a subgroup).
Take $G = D_4$, the dihedral group of order $8$ (also known as "the symmetries of the square"), let $r \in G$ be a rotation of $90^{\circ}$ (either direction), and $s$ any reflection in $D_4$(say, about the $x$-axis, for example).
Then $K = \{e,r^2,s,r^2s\}$ is a normal subgroup of $D_4$, being of index $2$.
Now $H = \{e,s\}$ is a non-normal subgroup of $G$, since $rsr^{-1} = rsr^3 = sr^{-1}r^3 = r^2 \not\in H$, and $H \cap K = H$.
(If $K$ is not required to be a proper subgroup of $G$, we can take $K = G = S_3$ and $H = \{e,(1\ 2)\}$).
