This section of the article focuses on finite group theory, specifically discussing Hall subgroups, Sylow bases, Carter subgroups, and the Fitting subgroup. 1. **Hall Subgroups and Sylow Bases**: - A subgroup \( H \) of a group \( G \) is a Hall \(\pi\)-subgroup if \( |H| \) is a product of primes in \(\pi\) and \( |G : H| \) is divisible by no prime in \(\pi\). A Hall \( p' \)-subgroup is called a \( p \)-complement. - Theorem 1 by P. Hall states that a group \( G \) is soluble if and only if it has a \( p \)-complement for every prime \( p \) dividing \( |G| \), has a Hall \(\pi\)-subgroup for every set of primes \(\pi\), and every \(\pi\)-subgroup of \( G \) is contained in a Hall \(\pi\)-subgroup, and all Hall \(\pi\)-subgroups are conjugate in \( G \). - A Sylow basis of \( G \) is a system of pairwise permutable Sylow subgroups, one for each prime factor of \( |G| \). - Theorem 2 states that a soluble group \( G \) possesses a Sylow basis, and all Sylow bases are conjugate in \( G \). - Corollary 3 provides conditions under which a subgroup \( H \) of a soluble group \( G \) reduces into a Sylow basis. 2. **Carter Subgroups**: - A Carter subgroup of a group \( G \) is a nilpotent subgroup \( C \) such that \( N_G(C) = C \). - Theorem 4 by R.W. Carter states that a soluble group \( G \) possesses Carter subgroups, and they are all conjugate. - Corollary 5 extends this to the case where \( G \) has a chain of normal subgroups, and the product of Carter subgroups of these subgroups forms \( G \). 3. **The Fitting Subgroup**: - The Fitting subgroup \(\text{Fit}(G)\) is the unique maximal nilpotent normal subgroup of \( G \). - A well-known fact is that in a finite soluble group, the Fitting subgroup contains its centralizer. - The section also includes a lemma that provides a sharper result about the Fitting subgroup.This section of the article focuses on finite group theory, specifically discussing Hall subgroups, Sylow bases, Carter subgroups, and the Fitting subgroup. 1. **Hall Subgroups and Sylow Bases**: - A subgroup \( H \) of a group \( G \) is a Hall \(\pi\)-subgroup if \( |H| \) is a product of primes in \(\pi\) and \( |G : H| \) is divisible by no prime in \(\pi\). A Hall \( p' \)-subgroup is called a \( p \)-complement. - Theorem 1 by P. Hall states that a group \( G \) is soluble if and only if it has a \( p \)-complement for every prime \( p \) dividing \( |G| \), has a Hall \(\pi\)-subgroup for every set of primes \(\pi\), and every \(\pi\)-subgroup of \( G \) is contained in a Hall \(\pi\)-subgroup, and all Hall \(\pi\)-subgroups are conjugate in \( G \). - A Sylow basis of \( G \) is a system of pairwise permutable Sylow subgroups, one for each prime factor of \( |G| \). - Theorem 2 states that a soluble group \( G \) possesses a Sylow basis, and all Sylow bases are conjugate in \( G \). - Corollary 3 provides conditions under which a subgroup \( H \) of a soluble group \( G \) reduces into a Sylow basis. 2. **Carter Subgroups**: - A Carter subgroup of a group \( G \) is a nilpotent subgroup \( C \) such that \( N_G(C) = C \). - Theorem 4 by R.W. Carter states that a soluble group \( G \) possesses Carter subgroups, and they are all conjugate. - Corollary 5 extends this to the case where \( G \) has a chain of normal subgroups, and the product of Carter subgroups of these subgroups forms \( G \). 3. **The Fitting Subgroup**: - The Fitting subgroup \(\text{Fit}(G)\) is the unique maximal nilpotent normal subgroup of \( G \). - A well-known fact is that in a finite soluble group, the Fitting subgroup contains its centralizer. - The section also includes a lemma that provides a sharper result about the Fitting subgroup.