Tools of algebra (3): Groups

a) A cyclic group $G$ is a group generated by a single element. Let $a \in G$ such that $\langle a \rangle = G$ , then $\forall \space b,c \in G, \space \exists \space m,n \in \mathbb{Z}$ such that $b = a^m$ and $c = a^n \Rightarrow b\cdot c = a^m \cdot a^n = a^n \cdot a^m = c \cdot b \Rightarrow$ G is an abelian group.

b) Each subgroup in an abelian group $G$ is a normal subgroup,then if $G$ is a simple abelian group and $G = \{e\}$ , $G$ is a cyclic group. If $G \neq \{e\}$ , take $a \in G$ with $a \neq e$ . Then the subroup generated by $a$ , thus $\langle a \rangle$ is a normal subgroup in $G$ and $\langle a \rangle \neq \{e\}$ and being $G$ a simple group, this implies that $\langle a \rangle = G$ , and therefore $G$ is a cyclic group.

c)

$c1) \Rightarrow c2)$ .- Apply Lagrange's theorem.

$c2) \Rightarrow c3)$ .- If $G$ is a finite group with $\{e\}$ and $G$ being the only subroups,we can take $a \neq e$ since $G \neq \{e\}$ and then $\langle a \rangle$ is a subgroup of $G\neq \{e\}\Rightarrow \langle a \rangle = G \Rightarrow$ $G$ is a cyclic group and being $G$ a finite group $G \cong \mathbb{Z}_p$ with $p$ a prime (applying Lagrange's theorem again)

$c3) \Rightarrow c1)$ .- Trivial