Let G be the additive group R × R and H = {(x, x) : x E R} be a subgroup of G. Give a geometric description of cosets of H. If N is a normal subgroup of order 2 of a group G then show that N CZ(G). If H is a subgroup of a group G such that (aH)(Hb) for any a, b e G is either a left or a right coset of H in G, prove that H is normal.

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