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.

Printable View

- April 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. - April 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 are the left cosets and the right then (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.

- April 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 - April 27th 2010, 02:35 PMrainyice
- April 27th 2010, 02:54 PMDrexel28
- April 28th 2010, 07:28 AMrainyice
- April 29th 2010, 08:12 AMMissMousey