Order of Group Element equals Order of Coprime Power/Corollary

Corollary to Order of Group Element equals Order of Coprime Power

Let $\struct {G, \circ}$ be a group whose identity is $e$.

Let $g \in G$ be an element of $g$.

Let $\order g$ denote the order of $g$ in $G$.

Let $H \le G$ be a subgroup of $G$.

Let $\order g = n$.

Let $g^m \in H$.

Let $m$ and $n$ be coprime.

Then $g \in H$.

Proof

From Order of Group Element equals Order of Coprime Power:

$m \perp n \iff \order {g^m} = \order g$

As $\order g = n$ and $n \perp m$, from Power of Generator of Cyclic Group is Generator iff Power is Coprime with Order:

$\gen g = \gen {g^m}$

Thus as $g^m \in H$ it follows that $\gen {g^m} \le H$

Hence $\gen g \le H$.

The result follows.

$\blacksquare$

Sources

• 1978: John S. Rose: A Course on Group Theory ... (previous) ... (next): $0$: Some Conventions and some Basic Facts: Exercise $12$