# Infinite Cyclic Groups Do Not Have Composition Series

## Problem 123

Let $G$ be an infinite cyclic group. Then show that $G$ does not have a composition series.

Contents

## Proof.

Let $G=\langle a \rangle$ and suppose that $G$ has a composition series
$G=G_0\rhd G_1 \rhd \cdots G_{m-1} \rhd G_m=\{e\},$ where $e$ is the identity element of $G$.

Note that each $G_i$ is an infinite cyclic subgroup of $G$.
Let $G_{m-1}=\langle b \rangle$. Then we have
$G_{m-1}=\langle b \rangle \rhd \langle b^2 \rangle \rhd \{e\}$ and the inclusions are proper.
(Since a cyclic group is abelian, these subgroups are normal in $G$.)

But this contradicts that $G_{m-1}$ is a simple group.
Thus, there is no composition series for an infinite cyclic group $G$.

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1. 09/27/2016

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##### Any Finite Group Has a Composition Series

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

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