# Tagged: fourth isomorphism theorem

## Problem 557

Let $N$ be a normal subgroup of a group $G$.

Suppose that $G/N$ is an infinite cyclic group.

Then prove that for each positive integer $n$, there exists a normal subgroup $H$ of $G$ of index $n$.

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Ring theory

by
Yu
· Published 11/24/2016
· Last modified 08/11/2017

## Problem 197

Let $R$ be a ring with unit $1\neq 0$.

Prove that if $M$ is an ideal of $R$ such that $R/M$ is a field, then $M$ is a maximal ideal of $R$.

(Do not assume that the ring $R$ is commutative.)

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Group Theory

by
Yu
· Published 09/26/2016
· Last modified 07/29/2017

## Problem 122

Let $G$ be a finite group. Then show that $G$ has a composition series.

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