is the intersection of subgroups a subgroup of each subgroup
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Suppose $G$ is a group, take $H,K$ as subgroups of $G$ so $H,Kleqslant G$ . I know that $Hcap Kleqslant G$ but is it the case that $Hcap Kleqslant H$ and $Hcap Kleqslant K$ ? I am guessing this does not hold but why? Also I tried with the case that $H=langle g rangle,K=langle h rangle$ where $g$ and $h$ are the elements in $G$ ( $langle h rangle$ means the minimum subgroup that contains the element $h$ if you haven't seen this notation before). I used the subspace test and I think that $Hcap Kleqslant H$ and $Hcap Kleqslant K$ hold unless I make a mistake somewhere. Much thanks in advance!
abstract-algebra group-theory
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