Pi-solvable group

A generalization of the concept of a solvable group. Let $\pi$ be a certain set of prime numbers. A finite group for which the order of each composition factor either is coprime to any member of $\pi$ or coincides with a certain prime in $\pi$, is called a $\pi$- solvable group. The basic properties of $\pi$- solvable groups are similar to the properties of solvable groups. A $\pi$- solvable group is a ${\pi }_1$- solvable group for any ${\pi }_1\subset \pi$; the subgroups, quotient groups and extensions of a $\pi$- solvable group by a $\pi$- solvable group are also $\pi$- solvable groups. In a $\pi$- solvable group $G$ every $\pi$- subgroup (that is, a subgroup all prime factors of the order of which belong to $\pi$) is contained in some Hall $\pi$- subgroup (a Hall $\pi$- subgroup is one with index in the group not divisible by any prime in $\pi$) and every ${\pi }^{\prime }$- subgroup (where ${\pi }^{\prime }$ is the complement of $\pi$ in the set of all prime numbers) is contained in some Hall ${\pi }^{\prime }$- subgroup; all Hall $\pi$- subgroups and also all Hall ${\pi }^{\prime }$- subgroups are conjugate in $G$; the index of a maximal subgroup of the group $G$ is either not divisible by any number in $\pi$ or is a power of one of the numbers of the set $\pi$( see [1]). The number of Hall $\pi$- subgroup in $G$ is equal to ${\alpha }_1\dots {\alpha }_t$, where ${\alpha }_i\equiv 1$( $\mathrm{mod}{p}_i$) for every $p_i\in \pi$ which divides the order of $G$, and, moreover, ${\alpha }_i$ divides the order of one of the chief factors of $G$( see [2]).

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