# $\pi$-groups and $\pi^{\prime}$-groups

Let $\pi$ be a set of primes. A torsion group $G$ is called a $\pi$-group if each prime dividing the order of an element of $G$ is in $\pi$ and a $\pi^{\prime}$-group if none of them are. Typically, if $\pi$ is a singleton $\pi=\{p\}$, we write $p$-group and $p^{\prime}$-group for these.

Remark. If $G$ is finite, then $G$ is a $\pi$-group if every prime dividing $|G|$ is in $\pi$.

Title $\pi$-groups and $\pi^{\prime}$-groups pigroupsAndpigroups 2013-03-22 13:17:51 2013-03-22 13:17:51 Algeboy (12884) Algeboy (12884) 9 Algeboy (12884) Definition msc 20D20 msc 20F50 $\pi$-group $\pi^{\prime}$-group