# If H is a subgroup of an abelian group G, then both H and the quotient group G/H are abelian

Hi! I'm using old tests to study for an abstract algebra exam and of course the old exams do not come with solutions. This means that I need help. I tend to struggle with proofs because I forget some steps or I am not as rigorous as I should be. Thus I need to see actual complete rigorous proofs so that I can make sure that I remember all of the necessary and sufficient steps. I would really appreciate the help. My exam is Wednesday.

I am submitting these questions exactly as they appear on the old exams.

First: prove

part a). If G/Z(G) is cyclic, then G is abelian.

part b). If Z(G) is maximal among abelian subgroups, then G is abelian.

Second:

Let |G| = infinity, and [G:H] < infinity. Show that H intersects every infinite subgroup of G nontrivially.

Third: prove or disprove

a). If H is a subgroup of an abelian group G, then both H and the

quotient group G/H are abelian

b). If H is a normal abelian subgroup of a group G, and the quotient

group G/H is also abelian, then G is abelian.

c). If K is a normal subgroup of H and H is a normal subgroup of G, then

K is a normal subgroup of G.

https://brainmass.com/math/basic-algebra/if-h-is-a-subgroup-of-an-abelian-group-g-then-both-h-and-the-quotient-group-g-h-are-abelian-223766

#### Solution Summary

This solution is comprised of a detailed explanation to answer if H is a subgroup of an abelian group G, then both H and the

quotient group G/H are abelian.