# Normal subgroup

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• July 30th 2008, 04:19 AM
mathemanyak
Normal subgroup
Let G be a group and let H<=G with [G:H]=2 show that H normal subgroup of G
• July 30th 2008, 09:52 AM
ThePerfectHacker
Quote:

Originally Posted by mathemanyak
Let G be a group and let H<=G with [G:H]=2 show that H normal subgroup of G

$H$ is a normal subgroup if and only if $aH=Ha$. Create the cosets $G/H$. If $a\in H$ then $aH=Ha$ because $G/H$ partitions the set $G$ into two sets. If $a\in G-H$ then $aH\not = H$ and $Ha\not = H$ thus $aH=Ha$ again because $G/H$ partitions the set into two sets.