On the pronormality and strong pronormality of Hall

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 H of G is called pronormal, if H and Hg are conjugate in ⟨H, Hg ⟩ for every g ∈ 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 ⟨H, Hg ⟩ for every g ∈ 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 ⟨H, Hg ⟩ for every g ∈ 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 ⟨H, Hg ⟩ for every g ∈ 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 ⟨H, Hg ⟩ for every g ∈ 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 ⟨H, Hg ⟩ for every g ∈ 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- (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 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 G = X ≀ Y there exists a nonpronormal 휋-Hall subgroup, whose normal closure is equal to G.

For example, {2, 3} satisfies conditions of the theorem: group X = PSL3(2) contains two classes of conjugated {2, 3}-Hall subgroups and group Y = PSL2(16) contains {2, 3}-Hall subgroup which is not equal to its normalizer in Y .Hence, in G = PSL3(2) ≀ PSL2(16) there exists a nonpronormal {2, 3}-Hall subgroup, whose normal closure is equal to G.

In “Kourovka Notebook” the next problem is formulated (see 18.32): is every Hall subgroup of a pronormal in its normal closure?

Nesterov Mikhail (NSU) Pronormality of Hall subgroups 2015 6 / 9 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 ≀ Y there exists a nonpronormal 휋-Hall subgroup, whose normal closure is equal to G.

For example, {2, 3} satisfies conditions of the theorem: group X = PSL3(2) contains two classes of conjugated {2, 3}-Hall subgroups and group Y = PSL2(16) contains {2, 3}-Hall subgroup which is not equal to its normalizer in Y .Hence, in G = PSL3(2) ≀ PSL2(16) there exists a nonpronormal {2, 3}-Hall subgroup, whose normal closure is equal to G.

In “Kourovka Notebook” the next problem is formulated (see 18.32): is every Hall subgroup of a finite group pronormal in its normal closure? The following theorem provides a negative answer to the problem

Theorem

Nesterov Mikhail (NSU) Pronormality of Hall subgroups 2015 6 / 9 (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 ≀ Y there exists a nonpronormal 휋-Hall subgroup, whose normal closure is equal to G.

For example, {2, 3} satisfies conditions of the theorem: group X = PSL3(2) contains two classes of conjugated {2, 3}-Hall subgroups and group Y = PSL2(16) contains {2, 3}-Hall subgroup which is not equal to its normalizer in Y .Hence, in G = PSL3(2) ≀ PSL2(16) there exists a nonpronormal {2, 3}-Hall subgroup, whose normal closure is equal to G.

In “Kourovka Notebook” the next problem is formulated (see 18.32): is every Hall subgroup of a finite group pronormal in its normal closure? The following theorem provides a negative answer to the problem

Theorem Let 휋 be such that

Nesterov Mikhail (NSU) Pronormality of Hall subgroups 2015 6 / 9 (2) there exists a simple group Y containing a non self-normalizing 휋-Hall subgroup. Thus in the regular wreath product G = X ≀ Y there exists a nonpronormal 휋-Hall subgroup, whose normal closure is equal to G.

For example, {2, 3} satisfies conditions of the theorem: group X = PSL3(2) contains two classes of conjugated {2, 3}-Hall subgroups and group Y = PSL2(16) contains {2, 3}-Hall subgroup which is not equal to its normalizer in Y .Hence, in G = PSL3(2) ≀ PSL2(16) there exists a nonpronormal {2, 3}-Hall subgroup, whose normal closure is equal to G.

In “Kourovka Notebook” the next problem is formulated (see 18.32): is every Hall subgroup of a finite group pronormal in its normal closure? 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;

Nesterov Mikhail (NSU) Pronormality of Hall subgroups 2015 6 / 9 Thus in the regular wreath product G = X ≀ Y there exists a nonpronormal 휋-Hall subgroup, whose normal closure is equal to G.

For example, {2, 3} satisfies conditions of the theorem: group X = PSL3(2) contains two classes of conjugated {2, 3}-Hall subgroups and group Y = PSL2(16) contains {2, 3}-Hall subgroup which is not equal to its normalizer in Y .Hence, in G = PSL3(2) ≀ PSL2(16) there exists a nonpronormal {2, 3}-Hall subgroup, whose normal closure is equal to G.

In “Kourovka Notebook” the next problem is formulated (see 18.32): is every Hall subgroup of a finite group pronormal in its normal closure? 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.

Nesterov Mikhail (NSU) Pronormality of Hall subgroups 2015 6 / 9 For example, {2, 3} satisfies conditions of the theorem: group X = PSL3(2) contains two classes of conjugated {2, 3}-Hall subgroups and group Y = PSL2(16) contains {2, 3}-Hall subgroup which is not equal to its normalizer in Y .Hence, in G = PSL3(2) ≀ PSL2(16) there exists a nonpronormal {2, 3}-Hall subgroup, whose normal closure is equal to G.

In “Kourovka Notebook” the next problem is formulated (see 18.32): is every Hall subgroup of a finite group pronormal in its normal closure? 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 ≀ Y there exists a nonpronormal 휋-Hall subgroup, whose normal closure is equal to G.

Nesterov Mikhail (NSU) Pronormality of Hall subgroups 2015 6 / 9 group X = PSL3(2) contains two classes of conjugated {2, 3}-Hall subgroups and group Y = PSL2(16) contains {2, 3}-Hall subgroup which is not equal to its normalizer in Y .Hence, in G = PSL3(2) ≀ PSL2(16) there exists a nonpronormal {2, 3}-Hall subgroup, whose normal closure is equal to G.

In “Kourovka Notebook” the next problem is formulated (see 18.32): is every Hall subgroup of a finite group pronormal in its normal closure? 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 ≀ Y there exists a nonpronormal 휋-Hall subgroup, whose normal closure is equal to G.

For example, {2, 3} satisfies conditions of the theorem:

Nesterov Mikhail (NSU) Pronormality of Hall subgroups 2015 6 / 9 and group Y = PSL2(16) contains {2, 3}-Hall subgroup which is not equal to its normalizer in Y .Hence, in G = PSL3(2) ≀ PSL2(16) there exists a nonpronormal {2, 3}-Hall subgroup, whose normal closure is equal to G.

In “Kourovka Notebook” the next problem is formulated (see 18.32): is every Hall subgroup of a finite group pronormal in its normal closure? 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 ≀ Y there exists a nonpronormal 휋-Hall subgroup, whose normal closure is equal to G.

For example, {2, 3} satisfies conditions of the theorem: group X = PSL3(2) contains two classes of conjugated {2, 3}-Hall subgroups

Nesterov Mikhail (NSU) Pronormality of Hall subgroups 2015 6 / 9 Hence, in G = PSL3(2) ≀ PSL2(16) there exists a nonpronormal {2, 3}-Hall subgroup, whose normal closure is equal to G.

In “Kourovka Notebook” the next problem is formulated (see 18.32): is every Hall subgroup of a finite group pronormal in its normal closure? 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 ≀ Y there exists a nonpronormal 휋-Hall subgroup, whose normal closure is equal to G.

For example, {2, 3} satisfies conditions of the theorem: group X = PSL3(2) contains two classes of conjugated {2, 3}-Hall subgroups and group Y = PSL2(16) contains {2, 3}-Hall subgroup which is not equal to its normalizer in Y .

Nesterov Mikhail (NSU) Pronormality of Hall subgroups 2015 6 / 9 In “Kourovka Notebook” the next problem is formulated (see 18.32): is every Hall subgroup of a finite group pronormal in its normal closure? 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 ≀ Y there exists a nonpronormal 휋-Hall subgroup, whose normal closure is equal to G.

For example, {2, 3} satisfies conditions of the theorem: group X = PSL3(2) contains two classes of conjugated {2, 3}-Hall subgroups and group Y = PSL2(16) contains {2, 3}-Hall subgroup which is not equal to its normalizer in Y .Hence, in G = PSL3(2) ≀ PSL2(16) there exists a nonpronormal {2, 3}-Hall subgroup, whose normal closure is equal to G.

Nesterov Mikhail (NSU) Pronormality of Hall subgroups 2015 6 / 9 Problem II. Strong pronormality Hall subgroups of simple groups

Nesterov Mikhail (NSU) Pronormality of Hall subgroups 2015 7 / 9 Clearly, every strongly pronormal subgroup is pronormal. The classical examples of strongly pronormal subgroups are: ∙ normal subgroups; ∙ maximal subgroups; ∙ Sylow subgroups; ∙ Hall subgroups of solvable groups. There are examples pronormal, but not strong pronormal subgroups. N.Ch.Manzaeva in 2014 showed that if (for fixed 휋) to 휋-subgroups of G performed complete analog of Sylow’s theorem, the 휋-Hall subgroups in G are strongly pronormal. We also provide a negative answer to Problem 17.45(b): in a finite simple group, are Hall subgroups always strongly pronormal?

More specifically, it was shown that PSp10(7) contains a {2, 3}-Hall subgroup, which is not strongly pronormal. Note that there are not known examples of pronormal Hall subgroups which are not strongly pronormal before.

A subgroup H of G is called strongly pronormal if, for each K ≤ H and g ∈ G, K g is conjugate with a subgroup of H (but not necessary with K) by an element from ⟨H, K g ⟩.

Nesterov Mikhail (NSU) Pronormality of Hall subgroups 2015 8 / 9 The classical examples of strongly pronormal subgroups are: ∙ normal subgroups; ∙ maximal subgroups; ∙ Sylow subgroups; ∙ Hall subgroups of solvable groups. There are examples pronormal, but not strong pronormal subgroups. N.Ch.Manzaeva in 2014 showed that if (for fixed 휋) to 휋-subgroups of G performed complete analog of Sylow’s theorem, the 휋-Hall subgroups in G are strongly pronormal. We also provide a negative answer to Problem 17.45(b): in a finite simple group, are Hall subgroups always strongly pronormal?

More specifically, it was shown that PSp10(7) contains a {2, 3}-Hall subgroup, which is not strongly pronormal. Note that there are not known examples of pronormal Hall subgroups which are not strongly pronormal before.

A subgroup H of G is called strongly pronormal if, for each K ≤ H and g ∈ G, K g is conjugate with a subgroup of H (but not necessary with K) by an element from ⟨H, K g ⟩. Clearly, every strongly pronormal subgroup is pronormal.

Nesterov Mikhail (NSU) Pronormality of Hall subgroups 2015 8 / 9 ∙ normal subgroups; ∙ maximal subgroups; ∙ Sylow subgroups; ∙ Hall subgroups of solvable groups. There are examples pronormal, but not strong pronormal subgroups. N.Ch.Manzaeva in 2014 showed that if (for fixed 휋) to 휋-subgroups of G performed complete analog of Sylow’s theorem, the 휋-Hall subgroups in G are strongly pronormal. We also provide a negative answer to Problem 17.45(b): in a finite simple group, are Hall subgroups always strongly pronormal?

More specifically, it was shown that PSp10(7) contains a {2, 3}-Hall subgroup, which is not strongly pronormal. Note that there are not known examples of pronormal Hall subgroups which are not strongly pronormal before.

A subgroup H of G is called strongly pronormal if, for each K ≤ H and g ∈ G, K g is conjugate with a subgroup of H (but not necessary with K) by an element from ⟨H, K g ⟩. Clearly, every strongly pronormal subgroup is pronormal. The classical examples of strongly pronormal subgroups are:

Nesterov Mikhail (NSU) Pronormality of Hall subgroups 2015 8 / 9 ∙ maximal subgroups; ∙ Sylow subgroups; ∙ Hall subgroups of solvable groups. There are examples pronormal, but not strong pronormal subgroups. N.Ch.Manzaeva in 2014 showed that if (for fixed 휋) to 휋-subgroups of G performed complete analog of Sylow’s theorem, the 휋-Hall subgroups in G are strongly pronormal. We also provide a negative answer to Problem 17.45(b): in a finite simple group, are Hall subgroups always strongly pronormal?

More specifically, it was shown that PSp10(7) contains a {2, 3}-Hall subgroup, which is not strongly pronormal. Note that there are not known examples of pronormal Hall subgroups which are not strongly pronormal before.

A subgroup H of G is called strongly pronormal if, for each K ≤ H and g ∈ G, K g is conjugate with a subgroup of H (but not necessary with K) by an element from ⟨H, K g ⟩. Clearly, every strongly pronormal subgroup is pronormal. The classical examples of strongly pronormal subgroups are: ∙ normal subgroups;

Nesterov Mikhail (NSU) Pronormality of Hall subgroups 2015 8 / 9 ∙ Sylow subgroups; ∙ Hall subgroups of solvable groups. There are examples pronormal, but not strong pronormal subgroups. N.Ch.Manzaeva in 2014 showed that if (for fixed 휋) to 휋-subgroups of G performed complete analog of Sylow’s theorem, the 휋-Hall subgroups in G are strongly pronormal. We also provide a negative answer to Problem 17.45(b): in a finite simple group, are Hall subgroups always strongly pronormal?

More specifically, it was shown that PSp10(7) contains a {2, 3}-Hall subgroup, which is not strongly pronormal. Note that there are not known examples of pronormal Hall subgroups which are not strongly pronormal before.

A subgroup H of G is called strongly pronormal if, for each K ≤ H and g ∈ G, K g is conjugate with a subgroup of H (but not necessary with K) by an element from ⟨H, K g ⟩. Clearly, every strongly pronormal subgroup is pronormal. The classical examples of strongly pronormal subgroups are: ∙ normal subgroups; ∙ maximal subgroups;

Nesterov Mikhail (NSU) Pronormality of Hall subgroups 2015 8 / 9 ∙ Hall subgroups of solvable groups. There are examples pronormal, but not strong pronormal subgroups. N.Ch.Manzaeva in 2014 showed that if (for fixed 휋) to 휋-subgroups of G performed complete analog of Sylow’s theorem, the 휋-Hall subgroups in G are strongly pronormal. We also provide a negative answer to Problem 17.45(b): in a finite simple group, are Hall subgroups always strongly pronormal?

More specifically, it was shown that PSp10(7) contains a {2, 3}-Hall subgroup, which is not strongly pronormal. Note that there are not known examples of pronormal Hall subgroups which are not strongly pronormal before.

A subgroup H of G is called strongly pronormal if, for each K ≤ H and g ∈ G, K g is conjugate with a subgroup of H (but not necessary with K) by an element from ⟨H, K g ⟩. Clearly, every strongly pronormal subgroup is pronormal. The classical examples of strongly pronormal subgroups are: ∙ normal subgroups; ∙ maximal subgroups; ∙ Sylow subgroups;

Nesterov Mikhail (NSU) Pronormality of Hall subgroups 2015 8 / 9 N.Ch.Manzaeva in 2014 showed that if (for fixed 휋) to 휋-subgroups of G performed complete analog of Sylow’s theorem, the 휋-Hall subgroups in G are strongly pronormal. We also provide a negative answer to Problem 17.45(b): in a finite simple group, are Hall subgroups always strongly pronormal?

More specifically, it was shown that PSp10(7) contains a {2, 3}-Hall subgroup, which is not strongly pronormal. Note that there are not known examples of pronormal Hall subgroups which are not strongly pronormal before.

A subgroup H of G is called strongly pronormal if, for each K ≤ H and g ∈ G, K g is conjugate with a subgroup of H (but not necessary with K) by an element from ⟨H, K g ⟩. Clearly, every strongly pronormal subgroup is pronormal. The classical examples of strongly pronormal subgroups are: ∙ normal subgroups; ∙ maximal subgroups; ∙ Sylow subgroups; ∙ Hall subgroups of solvable groups. There are examples pronormal, but not strong pronormal subgroups.

Nesterov Mikhail (NSU) Pronormality of Hall subgroups 2015 8 / 9 We also provide a negative answer to Problem 17.45(b): in a finite simple group, are Hall subgroups always strongly pronormal?

More specifically, it was shown that PSp10(7) contains a {2, 3}-Hall subgroup, which is not strongly pronormal. Note that there are not known examples of pronormal Hall subgroups which are not strongly pronormal before.

A subgroup H of G is called strongly pronormal if, for each K ≤ H and g ∈ G, K g is conjugate with a subgroup of H (but not necessary with K) by an element from ⟨H, K g ⟩. Clearly, every strongly pronormal subgroup is pronormal. The classical examples of strongly pronormal subgroups are: ∙ normal subgroups; ∙ maximal subgroups; ∙ Sylow subgroups; ∙ Hall subgroups of solvable groups. There are examples pronormal, but not strong pronormal subgroups. N.Ch.Manzaeva in 2014 showed that if (for fixed 휋) to 휋-subgroups of G performed complete analog of Sylow’s theorem, the 휋-Hall subgroups in G are strongly pronormal.

Nesterov Mikhail (NSU) Pronormality of Hall subgroups 2015 8 / 9 More specifically, it was shown that PSp10(7) contains a {2, 3}-Hall subgroup, which is not strongly pronormal. Note that there are not known examples of pronormal Hall subgroups which are not strongly pronormal before.

A subgroup H of G is called strongly pronormal if, for each K ≤ H and g ∈ G, K g is conjugate with a subgroup of H (but not necessary with K) by an element from ⟨H, K g ⟩. Clearly, every strongly pronormal subgroup is pronormal. The classical examples of strongly pronormal subgroups are: ∙ normal subgroups; ∙ maximal subgroups; ∙ Sylow subgroups; ∙ Hall subgroups of solvable groups. There are examples pronormal, but not strong pronormal subgroups. N.Ch.Manzaeva in 2014 showed that if (for fixed 휋) to 휋-subgroups of G performed complete analog of Sylow’s theorem, the 휋-Hall subgroups in G are strongly pronormal. We also provide a negative answer to Problem 17.45(b): in a finite simple group, are Hall subgroups always strongly pronormal?

Nesterov Mikhail (NSU) Pronormality of Hall subgroups 2015 8 / 9 Note that there are not known examples of pronormal Hall subgroups which are not strongly pronormal before.

A subgroup H of G is called strongly pronormal if, for each K ≤ H and g ∈ G, K g is conjugate with a subgroup of H (but not necessary with K) by an element from ⟨H, K g ⟩. Clearly, every strongly pronormal subgroup is pronormal. The classical examples of strongly pronormal subgroups are: ∙ normal subgroups; ∙ maximal subgroups; ∙ Sylow subgroups; ∙ Hall subgroups of solvable groups. There are examples pronormal, but not strong pronormal subgroups. N.Ch.Manzaeva in 2014 showed that if (for fixed 휋) to 휋-subgroups of G performed complete analog of Sylow’s theorem, the 휋-Hall subgroups in G are strongly pronormal. We also provide a negative answer to Problem 17.45(b): in a finite simple group, are Hall subgroups always strongly pronormal?

More specifically, it was shown that PSp10(7) contains a {2, 3}-Hall subgroup, which is not strongly pronormal.

Nesterov Mikhail (NSU) Pronormality of Hall subgroups 2015 8 / 9 A subgroup H of G is called strongly pronormal if, for each K ≤ H and g ∈ G, K g is conjugate with a subgroup of H (but not necessary with K) by an element from ⟨H, K g ⟩. Clearly, every strongly pronormal subgroup is pronormal. The classical examples of strongly pronormal subgroups are: ∙ normal subgroups; ∙ maximal subgroups; ∙ Sylow subgroups; ∙ Hall subgroups of solvable groups. There are examples pronormal, but not strong pronormal subgroups. N.Ch.Manzaeva in 2014 showed that if (for fixed 휋) to 휋-subgroups of G performed complete analog of Sylow’s theorem, the 휋-Hall subgroups in G are strongly pronormal. We also provide a negative answer to Problem 17.45(b): in a finite simple group, are Hall subgroups always strongly pronormal?

More specifically, it was shown that PSp10(7) contains a {2, 3}-Hall subgroup, which is not strongly pronormal. Note that there are not known examples of pronormal Hall subgroups which are not strongly pronormal before. Nesterov Mikhail (NSU) Pronormality of Hall subgroups 2015 8 / 9 Thank for attention!

Nesterov Mikhail (NSU) Pronormality of Hall subgroups 2015 9 / 9