Proposition 1

Let $G$ be a group, let $a\in G$, and suppose that the order of $a$ is $m$. Prove that:

(1) $a^n=e$ if and only if $m\mid n$.

(2) $a^k=a^h$ if and only if $k\equiv h \pmod m$.

Criterion 1: The Definition of a Group

Let $G$ be a nonempty set, and suppose a binary operation “$\cdot$” is defined on $G$. If this binary operation satisfies the following conditions:

(1) Closure: $ \forall x,y \in G,\ x\cdot y\in G $

(2) Associativity: $ \forall x,y,z\in G,\ (x\cdot y)\cdot z=x\cdot (y\cdot z) $

(3) Identity element: $ \exists e\in G,\ \forall a\in G,\ e\cdot a=a\cdot e=a $

(4) Inverse element: $ \forall a\in G,\ \exists b\in G,\ a\cdot b=b\cdot a=e $

then $G$ is called a group with respect to “$\cdot$”.