Every subgroup of index 2 is normal?

[Oster]

I have to prove that every subgroup H of a group G with index(number of distinct cosets of the subgroup) 2 is normal.

I don't know how to start :'(

Please help.

 

[micromass]

Hi Oster!

You must prove that gN=Ng for all g. But what exactly are the cosets of N??

 

[Oster]

If the element g comes from H then both gH and Hg are equal to H.
If g comes from H complement then i know it must represent the "other" coset. Since cosets partition G, i know both these cosets must be equal to H complement.
More or less correct?

 

[micromass]

Yes, that's good!