Suppose G and H are groups and phi: G --> H is a homomorphism. Prove if A is a subgroup of G, then phi(A) is a subgroup of H and if B is a subgroup of H, then phi^(-1)(B) is a subgroup of G.
The first part is easily proved with: Phi(a)*Phi(b)^-1=Phi(ab^-1)
For the second one, suppose a is an element of Phi^-1(B). That means Phi(a)^-1 belongs to B => Phi(a^-1) belongs to B => a^-1 is an element of Phi^-1(B). It is non-empty because it contains the identity (why?). Associativity is inherited from G.
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