$\pi$-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
Canonical name	pigroupsAndpigroups
Date of creation	2013-03-22 13:17:51
Last modified on	2013-03-22 13:17:51
Owner	Algeboy (12884)
Last modified by	Algeboy (12884)
Numerical id	9
Author	Algeboy (12884)
Entry type	Definition
Classification	msc 20D20
Classification	msc 20F50
Defines	$\pi$-group
Defines	${\pi }^{\prime }$-group