Suppose G is a group with subgroup H Prove that if G H 2 th

Suppose G is a group with subgroup H. Prove that if [G: H] = 2, then gH = Hg for any g elementof G

Solution

Since [G : H] = 2, H G. Let g G \\ H. Then H and gH are two disjoint right cosets of H, while H and Hg are two disjoint left cosets of H. Further, since [G : H] = 2, we have H gH = G = H Hg. Now, since H and gH are disjoint and since H and Hg are also disjoint, hence gH = G\\H = Hg. Now, let g G. If g H, then gH = H and H = Hg so that gH = Hg.   If g H, then also, gH = Hg. Hence, in any case gH = Hg.