Let A, B, and C be groups, let alpha, beta, and gamma be homomorphisms with gamma times alpha = beta
alpha gamma beta
If alpha is surjective, prove that ker(gamma)= alpha((ker(beta)).
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© BrainMass Inc. brainmass.com October 9, 2019, 10:13 pm ad1c9bdddf
Please see the attachment.
From the condition, we have the following relations.
We have .
Now suppose is surjective, we want to show that
For any , we have , then . ...
This contains proofs regarding a surjective homomorphism and a normal subgroup.