Let G be a group and H a subgroup. Prove if [G:H]=2, then H is normal.

[mathmajor2013]

Let G be a group and H be a subgroup of G. Prove that if [G]=2, then H is a normal subgroup of G.

 

[micromass]

The same as in my other reply:

1) this should belong in the homework forums
2) what did you try?

 

[mathmajor2013]

I'm lost on this one. It doesn't make sense how the number of left cosets corresponds to the normality. #gH=#Hg doesn't seem like it necessarily means that gH=Hg.

 

[micromass]

That [G]=2 means that there are only two left cosets of H. Also, it means that there are only two right cosets of H: H and G\H.

Thus gH is H or G\H, and for Hg thesame thing. Does this help you?

 

[mathmajor2013]

I thought that G/H was the set of left or right cosets, not a coset itself? But yes that does help, thank you!

 

[micromass]

No, I mean G\H, not G/H. With G\H, I mean the set-theoretic difference, i.e. everything in G which is not in H.

 

[mathmajor2013]

Oh I see. Yes! Thank you.

 

[berkeman]

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