P-Torsion Subgroup

definitiongroup-theory

Definition

For prime $p$ and commutative group $G$.

$$G[p]=\{g\in G:{\mathrm{ord}}_{}(g)=p^k\text{ for some k that may depend on g }\}$$

Lemmas

1. $G[p]\le G$
2. $G[p]\ne \{e_Q\}⟺p\mid |G|$

Fact

$g,g^{\prime }\in G$, $g{g}^{\prime }=g^{\prime }g$, then ${\mathrm{ord}}_{}(g{g}^{\prime })\mid \mathrm{lcm}({\mathrm{ord}}_{}(g),{\mathrm{ord}}_{}(g^{\prime }))$.