PlanetMath (more info)
 Math for the people, by the people. Sponsor PlanetMath
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
Owner confidence rating: High Entry average rating: No information on entry rating
$\pi$-groups and $\pi'$-groups (Definition)

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 $\pi'$ -group if none of them are. Typically, if $\pi$ is a singleton $\pi=\{p\}$ , we write $p$ -group and $p'$ -group for these.

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




"$\pi$-groups and $\pi'$-groups" is owned by Algeboy. [ full author list (3) | owner history (2) ]
(view preamble | get metadata)

View style:

Also defines:  $\pi$-group, $\pi'$-group
Log in to rate this entry.
(view current ratings)

Cross-references: finite, singleton, order, torsion group, primes
There are 3 references to this entry.

This is version 6 of $\pi$-groups and $\pi'$-groups, born on 2002-12-20, modified 2009-02-20.
Object id is 3797, canonical name is PiGroupsAndPiGroups.
Accessed 4602 times total.

Classification:
AMS MSC20D20 (Group theory and generalizations :: Abstract finite groups :: Sylow subgroups, Sylow properties, $\pi$-groups, $\pi$-structure)
 20F50 (Group theory and generalizations :: Special aspects of infinite or finite groups :: Periodic groups; locally finite groups)

Pending Errata and Addenda
None.
[ View all 5 ]
Discussion
Style: Expand: Order:
forum policy

No messages.

Interact
post | correct | update request | add derivation | add example | add (any)