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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.

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.

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