# $\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 |
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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 |