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 $G$.

A Group of Order $20$ is Solvable
Prove that a group of order $20$ is solvable.
Hint.
Show that a group of order $20$ has a unique normal $5$-Sylow subgroup by Sylow's theorem.
See the post summary of Sylow’s Theorem to review Sylow's theorem.
Proof.
Let $G$ be a group of order $20$. The […]

A Simple Abelian Group if and only if the Order is a Prime Number
Let $G$ be a group. (Do not assume that $G$ is a finite group.)
Prove that $G$ is a simple abelian group if and only if the order of $G$ is a prime number.
Definition.
A group $G$ is called simple if $G$ is a nontrivial group and the only normal subgroups of $G$ is […]

Any Finite Group Has a Composition Series
Let $G$ be a finite group. Then show that $G$ has a composition series.
Proof.
We prove the statement by induction on the order $|G|=n$ of the finite group.
When $n=1$, this is trivial.
Suppose that any finite group of order less than $n$ has a composition […]

The Number of Elements Satisfying $g^5=e$ in a Finite Group is Odd
Let $G$ be a finite group. Let $S$ be the set of elements $g$ such that $g^5=e$, where $e$ is the identity element in the group $G$.
Prove that the number of elements in $S$ is odd.
Proof.
Let $g\neq e$ be an element in the group $G$ such that $g^5=e$.
As […]

Group of Order $pq$ Has a Normal Sylow Subgroup and Solvable
Let $p, q$ be prime numbers such that $p>q$.
If a group $G$ has order $pq$, then show the followings.
(a) The group $G$ has a normal Sylow $p$-subgroup.
(b) The group $G$ is solvable.
Definition/Hint
For (a), apply Sylow's theorem. To review Sylow's theorem, […]

Any Subgroup of Index 2 in a Finite Group is Normal
Show that any subgroup of index $2$ in a group is a normal subgroup.
Hint.
Left (right) cosets partition the group into disjoint sets.
Consider both left and right cosets.
Proof.
Let $H$ be a subgroup of index $2$ in a group $G$.
Let $e \in G$ be the identity […]

Normal Subgroup Whose Order is Relatively Prime to Its Index
Let $G$ be a finite group and let $N$ be a normal subgroup of $G$.
Suppose that the order $n$ of $N$ is relatively prime to the index $|G:N|=m$.
(a) Prove that $N=\{a\in G \mid a^n=e\}$.
(b) Prove that $N=\{b^m \mid b\in G\}$.
Proof.
Note that as $n$ and […]

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