I'm currently studying for my exams and came across nd old exam exercise which i can't solve.
I want to prove that: Every Qoutientgroup of a Hyperabelian Group is again Hyperabelian. We defined Hyperabelian as: Every non trivial Qoutientgroup group of G has a nontrivial abelian normal subgroup, Than G is said to be hyperabelian It's obvious that this statement is true by it's definiton but i'm struggeling with the formal proof
I'm sure that the answer is constructed through the third Isomorphism theorem. By the isomorphism theorem every Normal subgroup of a Qoutient group G/K is of the form N/K so again a qoutient but to construct the next step we would need to take a Qoutient of a Qoutient group and i'm not sure if im allowed to do this without further information. Could someone provide a sketch for the proof or explain me which steps i have to take.
You want to show that any quotient of $G/K$ has a nontrivial normal abelian subgroup.
Use the isomorphism theorem: $(G/K)/(N/K)\cong G/N$. The right hand side of the isomorphism has a nontrivial normal abelian subgroup, hence the left hand side does as well.