Group Theory

A cyclic group $G$ is a group that can be generated by a single element $a$, so that every element in $G$ has the form $a^i$ for some integer $i$. We denote the cyclic group of order $n$ by ${\mathbb{Z}}_n$, since the additive group of ${\mathbb{Z}}_n$ is a cyclic group of order $n$.

Theorem: All subgroups of a cyclic group are cyclic. If $G=⟨a⟩$ is cyclic, then for every divisor $d$ of $\mid G\mid$ there exists exactly one subgroup of order $d$ which may be generated by $a^{\mid G\mid /d}$.

Proof: Let $\mid G\mid =dn$. Then $1,a^n,a^{2n},...,a^{(d-1)n}$ are distinct and form a cyclic subgroup $⟨a^n⟩$ of order $d$. Conversely, let $H=\{1,a_1,...,a_{d-1}$ be a subgroup of $G$ for some $d$ dividing $G$. Then for all $i$, $a_i=a^k$ for some $k$, and since every element has order dividing $\mid H\mid$, $a_i^d=a^{kd}=1$. Thus $kd=\mid G\mid m=ndm$ for some $m$, and we have $a_i=a^{nm}$ so each $a_i$ is in fact a power of $a^n$. From above this means it must be one of the $d$ subgroups already described.

Theorem: Every group of composite order has proper subgroups.

Proof: Let $G$ be a group of composite order, and let $1\ne a\in G$. Then if $⟨a⟩\ne G$ we are done, otherwise the subgroup $⟨a^d⟩\ne G$ for every divisor $d$ of $\mid G\mid$.