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.
Add to solve later Let $G$ be a non-abelian simple group. Let $D(G)=[G,G]$ be the commutator subgroup of $G$. Show that $G=D(G)$.
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Add to solve later Let $G$ be a finite group. Then show that $G$ has a composition series.
Add to solve later Let $G$ 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 $|X|!$.
Add to solve later (a) Show that if a group $G$ has the following order, then it is not simple.
(b) Show that if the order of a group $G$ is equal to an even perfect number then the group is not simple.
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