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$p$-group (Definition)

Primary groups

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 $ \vert G\vert$ is a power of $ p$.

Primary subgroups

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.



"$p$-group" is owned by yark.
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See Also: $p$-subgroup, $p$-extension, pro-$p$ 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 10 references to this entry.

This is version 10 of $p$-group, born on 2004-12-12, modified 2007-05-21.
Object id is 6565, canonical name is PGroup4.
Accessed 5146 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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