# Subgroups

Let G be a group with A and B subgroups of G. Prove that the set AB = {ab | a is in A and b is in B} is a subgroup of G if and only if AB = BA ( ie, for any a in A, b in B, there exist elements a_1 in A, b_1 in B such that ab = b_1a_1, and there exist elements a_2 in A, b_2 in B such that ba = a_2b_2)

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Let G be a group with A and B subgroups of G. Prove that the set AB={ab ┤| a∈A,b∈B} is a subgroup of G if and only if AB=BA (i.e. for any a∈A,b∈B, there exist elements a_1∈A, b_1∈B such that ab=b_1 a_1, and there exist elements a_2∈A, b_2∈B such that ba=a_2 b_2.

Proof: Let G be a ...

#### Solution Summary

Subgroups are examined closely.

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