Let G be an abelian group and H a subgroup For any element g

Let G be an abelian group and H a subgroup. For any element g of G, where g has order 2, define gH={ghhH}. Prove that the set K=HgH is a subgroup of G.

Solution

Note that g2=1 implies g=g1.

(a) 1H, so 1HgH

(b) Suppose x,yHgH.

There are four cases

1. x,yH: then xyH

2. xH, y=ghgH: then xy=g(xh)gH

3. x=ghgH, yH: then xy=g(hy)gH

4. x=gh1gH, y=gh2gH: then xy=g2h1h2=h1h2H

(c) If xH, then x1H; if x=ghgH, then x1=h1g1=g1h1=gh1gH