## A Simple Abelian Group if and only if the Order is a Prime Number

## Problem 290

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.

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.

Let $G$ be a non-abelian simple group. Let $D(G)=[G,G]$ be the commutator subgroup of $G$. Show that $G=D(G)$.

Read solution

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

Add to solve laterLet $G$ be a simple group and let $X$ be a finite set.

Suppose $G$ acts nontrivially on $X$. That is, there exist $g\in G$ and $x \in X$ such that $g\cdot x \neq x$.

Then show that $G$ is a finite group and the order of $G$ divides $|X|!$.

**(a)** Show that if a group $G$ has the following order, then it is not simple.

- $28$
- $496$
- $8128$

**(b) **Show that if the order of a group $G$ is equal to an even * perfect number* then the group is not simple.