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 dont 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!

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Oster	Every subgroup of index 2 is normal? Tweet 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 dont know how to start :'( Please help.

micromass Blog Entries: 8 Recognitions: Gold Member Science Advisor Staff Emeritus	Hi Oster! You must prove that gN=Ng for all g. But what exactly are the cosets of N??