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 -group
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(Definition)
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Let $p$ be a prime number. A $p$ group (or $p$ primary group) is a group in which the order of every element is a power of $p$ A group that is a $p$ group for some prime $p$ is also called a primary group.
Using Lagrange's Theorem and Cauchy's Theorem one may show that a finite group $G$ is a $p$ group if and only if $|G|$ is a power of $p$
A $p$ subgroup (or $p$ primary subgroup) of a group $G$ is a subgroup $H$ of $G$ such that $H$ is also a $p$ group. A group that is a $p$ subgroup for some prime $p$ is also called a primary subgroup.
It follows from Zorn's Lemma that every group has a maximal $p$ subgroup, for every prime $p$ The maximal $p$ subgroup need not be unique (though for abelian groups it is always unique, and is called the $p$ primary component of the abelian group). A maximal $p$ subgroup may, of course, be trivial. Non-trivial maximal $p$ subgroups of finite groups are called Sylow $p$ subgroups.
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" -group" is owned by yark.
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See Also:  -subgroup, -extension, pro- group, quasicyclic group, subgroup
| Other names: |
p-group, p-primary group, primary group |
| Also defines: |
p-subgroup, primary component, p-primary, p-primary subgroup, primary subgroup |
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Cross-references: abelian groups, Zorn's lemma, finite group, Cauchy's theorem, Lagrange's theorem, power, order, group, prime number
There are 11 references to this entry.
This is version 10 of -group, born on 2004-12-12, modified 2007-05-21.
Object id is 6565, canonical name is PGroup4.
Accessed 7874 times total.
Classification:
| AMS MSC: | 20F50 (Group theory and generalizations :: Special aspects of infinite or finite groups :: Periodic groups; locally finite groups) |
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Pending Errata and Addenda
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