# Groups and proofs

I)

Let A, B, and C be groups, let alpha, beta, and gamma be homomorphisms with gamma times alpha = beta

alpha gamma beta

A--------->B----------->C<---------A

If alpha is surjective, prove that ker(gamma)= alpha((ker(beta)).

ii)

Prove that if K is a subgroup of a group G, and if every left coset of aK is equal to the right coset Kb, then K is a normal subgroup of G.

I just dont know how to do these, the confuse me and I am horrible at writing them in good proof form

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#### Solution Preview

Please see the attachment.

Problem #1

Proof:

From the condition, we have the following relations.

,

We have .

Now suppose is surjective, we want to show that

For any , we have , then . ...

#### Solution Summary

This contains proofs regarding a surjective homomorphism and a normal subgroup.