Infinite Cyclic Groups Do Not Have Composition Series

Group Theory Problems and Solutions in Mathematics

Problem 123

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

 
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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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