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On the Pronormality and Strong Pronormality of Hall Subgroups

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On the Pronormality and Strong Pronormality of Hall Subgroups 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.
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