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$\pi$-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 $ \vert G\vert$ is in $ \pi$.



"$\pi$-groups and $\pi'$-groups" is owned by Algeboy. [ full author list (2) | owner history (2) ]
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Also defines:  $\pi$-group, $\pi'$-group
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Cross-references: finite, singleton, order, torsion group, primes
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This is version 5 of $\pi$-groups and $\pi'$-groups, born on 2002-12-20, modified 2007-12-07.
Object id is 3797, canonical name is PiGroupsAndPiGroups.
Accessed 3655 times total.

Classification:
AMS MSC: 20D20 (Group theory and generalizations :: Abstract finite groups :: Sylow subgroups, Sylow properties, $\pi$-groups, $\pi$-structure)
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