Let H be a group and K a subgroup of H with (H:K)=n< ∞.

Prove: if (H:K)=2, then K is a normal subgroup of H

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- Nov 14th 2009, 08:29 PMapple2009normal subgroup
Let H be a group and K a subgroup of H with (H:K)=n< ∞.

Prove: if (H:K)=2, then K is a normal subgroup of H - Nov 15th 2009, 12:32 AMOpalg
The subgroup K partitions H into left cosets, one of which is K itself. If (H:K)=2, then there is only one other left coset, namely everything in H that isn't in K. The same applies to right cosets. Thus when (H:K)=2, each left coset is also a right coset. But that is one way of expressing the condition for K to be normal.