Subgroup and quotient

Algebra Level 4

Let $G$ be a group, and $N$ a normal subgroup. Which of the following statements is/are always true?

I. If $N$ is finite and $G/N$ is finite, then $G$ is finite.
II. If $N$ is finite and cyclic and $G/N$ is finite and cyclic, then $G$ is finite and cyclic.
III. If $N$ is abelian and $G/N$ is abelian, then $G$ is abelian.

Notation:

A finite cyclic group is a group that is isomorphic to ${\mathbb Z}_n,$ the integers mod $n,$ for some $n.$
An abelian group is a group whose operation is commutative: $x * y = y * x$ for all $x,y \in G.$

I only III only II and III II only I and II I and III

1 solution

Patrick Corn
Jun 25, 2016

I is true: the elements of $G$ are partitioned into disjoint cosets $gN.$ Since $N$ is finite, the cosets are finite. Since $G/N$ is finite, there are finitely many cosets. So $G$ is finite (and indeed $|G| = |N| \cdot |G/N|$ ).

II and III are false. Here is a counterexample for both. Let $G = S_3,$ the symmetric group on three symbols, and $N = A_3 = \{id,(123),(132)\}.$ Then $N$ is cyclic (isomorphic to ${\mathbb Z}_3$ ) and $G/N$ is cyclic (isomorphic to ${\mathbb Z}_2$ ) but $G$ is neither cyclic nor abelian.

Subgroup and quotient

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