On the pronormality and strong pronormality of Hall subgroups Mikhail Nesterov Novosibirsk State University 2015 Nesterov Mikhail (NSU) Pronormality of Hall subgroups 2015 1 / 9 Introduction. Basic concepts Nesterov Mikhail (NSU) Pronormality of Hall subgroups 2015 2 / 9 The classical examples of pronormal subgroups are: ∙ normal subgroups; ∙ maximal subgroups; ∙ Sylow subgroups; ∙ Hall subgroups of solvable groups. A subgroup H of G is called pronormal, if H and Hg are conjugate in hH; Hg i for every g 2 G. Nesterov Mikhail (NSU) Pronormality of Hall subgroups 2015 3 / 9 ∙ normal subgroups; ∙ maximal subgroups; ∙ Sylow subgroups; ∙ Hall subgroups of solvable groups. A subgroup H of G is called pronormal, if H and Hg are conjugate in hH; Hg i for every g 2 G. The classical examples of pronormal subgroups are: Nesterov Mikhail (NSU) Pronormality of Hall subgroups 2015 3 / 9 ∙ maximal subgroups; ∙ Sylow subgroups; ∙ Hall subgroups of solvable groups. A subgroup H of G is called pronormal, if H and Hg are conjugate in hH; Hg i for every g 2 G. The classical examples of pronormal subgroups are: ∙ normal subgroups; Nesterov Mikhail (NSU) Pronormality of Hall subgroups 2015 3 / 9 ∙ Sylow subgroups; ∙ Hall subgroups of solvable groups. A subgroup H of G is called pronormal, if H and Hg are conjugate in hH; Hg i for every g 2 G. The classical examples of pronormal subgroups are: ∙ normal subgroups; ∙ maximal subgroups; Nesterov Mikhail (NSU) Pronormality of Hall subgroups 2015 3 / 9 ∙ Hall subgroups of solvable groups. A subgroup H of G is called pronormal, if H and Hg are conjugate in hH; Hg i for every g 2 G. The classical examples of pronormal subgroups are: ∙ normal subgroups; ∙ maximal subgroups; ∙ Sylow subgroups; Nesterov Mikhail (NSU) Pronormality of Hall subgroups 2015 3 / 9 A subgroup H of G is called pronormal, if H and Hg are conjugate in hH; Hg i for every g 2 G. The classical examples of pronormal subgroups are: ∙ normal subgroups; ∙ maximal subgroups; ∙ Sylow subgroups; ∙ Hall subgroups of solvable groups. Nesterov Mikhail (NSU) Pronormality of Hall subgroups 2015 3 / 9 Throughout a set of primes is denoted by 휋. A subgroup H of G is called a 휋-Hall subgroup, if H is a 휋-group (i.e. all its prime divisors are in 휋), while the index of H is not divisible by primes from 휋. A subgroup is said to be a Hall subgroup if it is a 휋-Hall subgroup for a set of primes 휋. Hall’s theorem states that a finite soluble group contain exactly one class of conjugated 휋-Hall subgroups for any set 휋 of primes. As a result, Hall subgroups in solvable groups pronormality. The insoluble group may be contain more than one class of conjugated 휋-Hall subgroups. The Hall subgroups may also be not pronormal. Recall the definition of Hall subgroups. Nesterov Mikhail (NSU) Pronormality of Hall subgroups 2015 4 / 9 A subgroup H of G is called a 휋-Hall subgroup, if H is a 휋-group (i.e. all its prime divisors are in 휋), while the index of H is not divisible by primes from 휋. A subgroup is said to be a Hall subgroup if it is a 휋-Hall subgroup for a set of primes 휋. Hall’s theorem states that a finite soluble group contain exactly one class of conjugated 휋-Hall subgroups for any set 휋 of primes. As a result, Hall subgroups in solvable groups pronormality. The insoluble group may be contain more than one class of conjugated 휋-Hall subgroups. The Hall subgroups may also be not pronormal. Recall the definition of Hall subgroups. Throughout a set of primes is denoted by 휋. Nesterov Mikhail (NSU) Pronormality of Hall subgroups 2015 4 / 9 A subgroup is said to be a Hall subgroup if it is a 휋-Hall subgroup for a set of primes 휋. Hall’s theorem states that a finite soluble group contain exactly one class of conjugated 휋-Hall subgroups for any set 휋 of primes. As a result, Hall subgroups in solvable groups pronormality. The insoluble group may be contain more than one class of conjugated 휋-Hall subgroups. The Hall subgroups may also be not pronormal. Recall the definition of Hall subgroups. Throughout a set of primes is denoted by 휋. A subgroup H of G is called a 휋-Hall subgroup, if H is a 휋-group (i.e. all its prime divisors are in 휋), while the index of H is not divisible by primes from 휋. Nesterov Mikhail (NSU) Pronormality of Hall subgroups 2015 4 / 9 Hall’s theorem states that a finite soluble group contain exactly one class of conjugated 휋-Hall subgroups for any set 휋 of primes. As a result, Hall subgroups in solvable groups pronormality. The insoluble group may be contain more than one class of conjugated 휋-Hall subgroups. The Hall subgroups may also be not pronormal. Recall the definition of Hall subgroups. Throughout a set of primes is denoted by 휋. A subgroup H of G is called a 휋-Hall subgroup, if H is a 휋-group (i.e. all its prime divisors are in 휋), while the index of H is not divisible by primes from 휋. A subgroup is said to be a Hall subgroup if it is a 휋-Hall subgroup for a set of primes 휋. Nesterov Mikhail (NSU) Pronormality of Hall subgroups 2015 4 / 9 As a result, Hall subgroups in solvable groups pronormality. The insoluble group may be contain more than one class of conjugated 휋-Hall subgroups. The Hall subgroups may also be not pronormal. Recall the definition of Hall subgroups. Throughout a set of primes is denoted by 휋. A subgroup H of G is called a 휋-Hall subgroup, if H is a 휋-group (i.e. all its prime divisors are in 휋), while the index of H is not divisible by primes from 휋. A subgroup is said to be a Hall subgroup if it is a 휋-Hall subgroup for a set of primes 휋. Hall’s theorem states that a finite soluble group contain exactly one class of conjugated 휋-Hall subgroups for any set 휋 of primes. Nesterov Mikhail (NSU) Pronormality of Hall subgroups 2015 4 / 9 The insoluble group may be contain more than one class of conjugated 휋-Hall subgroups. The Hall subgroups may also be not pronormal. Recall the definition of Hall subgroups. Throughout a set of primes is denoted by 휋. A subgroup H of G is called a 휋-Hall subgroup, if H is a 휋-group (i.e. all its prime divisors are in 휋), while the index of H is not divisible by primes from 휋. A subgroup is said to be a Hall subgroup if it is a 휋-Hall subgroup for a set of primes 휋. Hall’s theorem states that a finite soluble group contain exactly one class of conjugated 휋-Hall subgroups for any set 휋 of primes. As a result, Hall subgroups in solvable groups pronormality. Nesterov Mikhail (NSU) Pronormality of Hall subgroups 2015 4 / 9 The Hall subgroups may also be not pronormal. Recall the definition of Hall subgroups. Throughout a set of primes is denoted by 휋. A subgroup H of G is called a 휋-Hall subgroup, if H is a 휋-group (i.e. all its prime divisors are in 휋), while the index of H is not divisible by primes from 휋. A subgroup is said to be a Hall subgroup if it is a 휋-Hall subgroup for a set of primes 휋. Hall’s theorem states that a finite soluble group contain exactly one class of conjugated 휋-Hall subgroups for any set 휋 of primes. As a result, Hall subgroups in solvable groups pronormality. The insoluble group may be contain more than one class of conjugated 휋-Hall subgroups. Nesterov Mikhail (NSU) Pronormality of Hall subgroups 2015 4 / 9 Recall the definition of Hall subgroups. Throughout a set of primes is denoted by 휋. A subgroup H of G is called a 휋-Hall subgroup, if H is a 휋-group (i.e. all its prime divisors are in 휋), while the index of H is not divisible by primes from 휋. A subgroup is said to be a Hall subgroup if it is a 휋-Hall subgroup for a set of primes 휋. Hall’s theorem states that a finite soluble group contain exactly one class of conjugated 휋-Hall subgroups for any set 휋 of primes. As a result, Hall subgroups in solvable groups pronormality. The insoluble group may be contain more than one class of conjugated 휋-Hall subgroups. The Hall subgroups may also be not pronormal. Nesterov Mikhail (NSU) Pronormality of Hall subgroups 2015 4 / 9 Problem I. Pronormality of Hall subgroups in its normal closure Nesterov Mikhail (NSU) Pronormality of Hall subgroups 2015 5 / 9 The following theorem provides a negative answer to the problem Theorem Let 휋 be such that (1) there exists a simple group X containing more than one class of conjugated 휋-Hall subgroups; (2) there exists a simple group Y containing a non self-normalizing 휋-Hall subgroup. Thus in the regular wreath product G = X o Y there exists a nonpronormal 휋-Hall subgroup, whose normal closure is equal to G. For example, f2; 3g satisfies conditions of the theorem: group X = PSL3(2) contains two classes of conjugated f2; 3g-Hall subgroups and group Y = PSL2(16) contains f2; 3g-Hall subgroup which is not equal to its normalizer in Y .Hence, in G = PSL3(2) o PSL2(16) there exists a nonpronormal f2; 3g-Hall subgroup, whose normal closure is equal to G.