Let G be a group. For subsets A, B C G, define AB = {ab |...

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Let G be a group. For subsets A, B C G, define AB = {ab | a € A and b € B}. That is, AB consists of all group elements in G that can be obtained by choosing elements a € A, be B and multiplying them (using G's group operation). 1. Show: if H≤ G is a subgroup of G, then HH = H. 2. Show: if G is a finite group, HCG is nonempty, and HH = H, then H is a subgroup of H. 3. Show: there are an infinite group G and a nonempty subset HCG such that HH = H, but H is not a subgroup of G. (You show this last point by giving an example). Let G be a group. For subsets A, B C G, define AB = {ab | a € A and b € B}. That is, AB consists of all group elements in G that can be obtained by choosing elements a € A, be B and multiplying them (using G's group operation). 1. Show: if H≤ G is a subgroup of G, then HH = H. 2. Show: if G is a finite group, HCG is nonempty, and HH = H, then H is a subgroup of H. 3. Show: there are an infinite group G and a nonempty subset HCG such that HH = H, but H is not a subgroup of G. (You show this last point by giving an example).

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a Since AB is a subset of a group G because ab is in G so ab i... View the full answer