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Given a group G with a subgroup H, define a binary relation on G by a...

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Advance Math

Given a group G with a subgroup H, define a binary relation on Gby a ? b if and only if ba^(-1)? H.

(a) (5 points) Prove that ? is an equivalence relation.

(b) (5 points) For each a ? G denote by [a] the equivalenceclass of a and prove that [a] = Ha = {ha | h ? H}. A set of theform Ha, for some a ? G, is called a right coset of H in G.

(c) (5 points) Let a ? G. For all g ? G prove that Hg = Ha ifand only if g ? Ha. Hint: two elements are equivalent if and onlyif their equivalence classes coincide.

(d) (5 points) Prove that the map ?a : H ? Ha given by ?a(h) =ha, h ? H, is a bijection.

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