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$\pi$-separable group (Definition)

Let $ \pi$ be a set of prime numbers. A finite group $ G$ is called $ \pi$-separable if there exists a composition series

$\displaystyle \{1\}=G_0\lhd\cdots\lhd G_n=G $
such that each $ G_{i+1}/G_i$ is either a $ \pi$-group or a $ \pi'$-group.

A $ \{p\}$-separable group, where $ p$ is a prime number, is usually called a $ p$-separable group.

$ \pi$-separability can be thought of as a generalization of solvability for finite groups; a finite group is $ \pi$-separable for all sets of primes if and only it is solvable.



"$\pi$-separable group" is owned by yark. [ full author list (2) | owner history (1) ]
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Also defines:  $\pi$-separable, $p$-separable, $\pi$-separability, $p$-separability, $p$-separable group
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Cross-references: solvable, group, composition series, finite group, prime numbers
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This is version 4 of $\pi$-separable group, born on 2002-12-20, modified 2007-12-08.
Object id is 3796, canonical name is Seperable.
Accessed 4472 times total.

Classification:
AMS MSC: 20D10 (Group theory and generalizations :: Abstract finite groups :: Solvable groups, theory of formations, Schunck classes, Fitting classes, $\pi$-length, ranks)
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