Give an example to show that if A and B are subgroups of a group G, the AB = {ab
ID: 2980055 • Letter: G
Question
Give an example to show that if A and B are subgroups of a group G, the AB = {ab : a ? A and b ? B} need not be a subgroup of G.Explanation / Answer
A and B belong to G , .................................................................................................................................................. AB = { ab : a ?A and b?B } , .................................................................................................................................................. since A and B are subsets of G , .................................................................................................................................................. therefore, but for A>1 and B>1 , AB > A and AB>B , .................................................................................................................................................. let A be the maximum value of G, .................................................................................................................................................. and B be any value greater than 1, .................................................................................................................................................. then, AB would be having a value that is greater than maximum value of G, .................................................................................................................................................. therefore, in that case AB would not belong to G ................................................................................................................................................... therefore , AB need not be a subgroup of G.
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