# Math Help - plz solve these problems.....

1. ## plz solve these problems.....

1:Show that union of two subgroups may not be a group?

2:If G is a group and H is a subgroup of index 2 in G,prove that H is normal subgroup of G?

2. Originally Posted by Mathventure
1:Show that union of two subgroups may not be a group?
Take $S_3$ and $<(12)>$ with $<(13)>$.

2:If G is a group and H is a subgroup of index 2 in G,prove that H is normal subgroup of G?
Let $a\in G$. If $a\in H$ then $aH=Ha$. If $a\not \in H$ then $aH$ is the coset other than $H$, and $Ha$ is the coset other than $H$. Since there are only two cosets it means $aH=Ha$. Thus, $H$ is a normal subgroup of $G$.