Let G be a group. ACG to be we have defined the centralizer of a subset...

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Let G be a group. ACG to be we have defined the centralizer of a subset C(A) = {G| for every y € A, ry = yr} the set of group elements that commute with every element of A. We have seen that C(A) is a subgroup of G. Let's define for two subsets A. BCG: AB= (ab a A and be B). So AB is the set of all group elements that can be obtained by picking a member a of A, a member b of B, and multiplying them (using G's group operation). Now suppose HSG is a subgroup of G. Your problem is to show that HC(H)<G that is, that HC(H) is a subgroup of G. Let G be a group. ACG to be we have defined the centralizer of a subset C(A) = {G| for every y € A, ry = yr} the set of group elements that commute with every element of A. We have seen that C(A) is a subgroup of G. Let's define for two subsets A. BCG: AB= (ab a A and be B). So AB is the set of all group elements that can be obtained by picking a member a of A, a member b of B, and multiplying them (using G's group operation). Now suppose HSG is a subgroup of G. Your problem is to show that HC(H)<G that is, that HC(H) is a subgroup of G.

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Sol 1 ii HCCHI ha hH a CCH Z 6 hia hb HCH where then ie then 6t ... View the full answer