Show that if K≤Gand(G:K)=2,then gK=Kg for every g∈G.

Question

This is a question on my abstract algebra hw. It’s about cosets and index but i have no idea what to do

Answer 1

If g ∈ K then gK = K = Kg. Now suppose g ∈ G - K. Then gK (and Kg) are different from K. Since (G : K) = 2, K and gK are the left cosets; K and Kg are the right cosets. Left cosets and right cosets partition G, so gK = G - K = Kg.