Let H be a subgroup of G with index 2.

a. Prove that H is a normal subgroup of G.

b. Prove that g^(2) E H for all g E G.

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- Apr 27th 2010, 11:55 AMrainyiceProve normal subgroup
Let H be a subgroup of G with index 2.

a. Prove that H is a normal subgroup of G.

b. Prove that g^(2) E H for all g E G. - Apr 27th 2010, 12:01 PMDrexel28
There is a fairly well-known theorem which says that "if all the right cosets of a subgroup are left cosets of the subgroup then the subgroup is normal". So, notice that if $\displaystyle L_1,L_2$ are the left cosets and $\displaystyle R_1,R_2$ the right then $\displaystyle L_1\amalg L_2=R_1\amalg R_2$ (where that funny symbol just means union but the sets are pairwise disjoint). What now?

Quote:

b. Prove that g^(2) E H for all g E G.

- Apr 27th 2010, 01:22 PMglasssocks
For the first part:

for g E G

if g E H , then gH = H = Hg. Therefore H is a normal subgroup.

if g is not an element of H, G = eH U gH because there are only 2 cosets.

also G = He U Hg.

So G = H U Hg and G = H U gH (H = eH = He)

Therefore, gH = Hg. H is a normal subgroup.

I don't know how to prove part b. Sorry - Apr 27th 2010, 02:35 PMrainyice
- Apr 27th 2010, 02:54 PMDrexel28
That's lame. I wrote up a proof. Here's the basic idea. Break it into two cases

1. If $\displaystyle g\in N$ it's obvious what to do.

2. If $\displaystyle g\notin N$ then it has to be in the other coset, call it $\displaystyle hN$. And, so if $\displaystyle g^2\notin N\implies\cdots g\notin g^{-1}N\implies\cdots$ - Apr 28th 2010, 07:28 AMrainyice
- Apr 29th 2010, 08:12 AMMissMousey