Baer-Suzuki theorem for the π-radical

Yang, Nanying; Ревин, Д. О.; Vdovin, E. P.

doi:10.1007/s11856-021-2209-y

Cited by 9 publications

(13 citation statements)

References 24 publications

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“…At the same time, this width is finite for any π, 3 and the following question arises: what is the precise value of the width? A conjecture has been made (see [32], Conjecture 1) that BS(π) in most cases coincides with the lower bound r − 1. More precisely, there is a question of whether or not the following assertion holds.…”

Section: § 1 Introductionmentioning

confidence: 99%

“…In [32], Theorem 1.2, the authors proved that for any π ⊆ P there always exists a nonnegative integer m = m(π) such that for an arbitrary group G O π (G) = {x ∈ G | ⟨x g1 , . .…”

Section: § 1 Introductionmentioning

confidence: 99%

“…By Definition 1.15 in [15] the least m of this kind is called the Baer-Suzuki width of the class of π-groups and is denoted by BS(π). Moreover, it was proved (see [32], Theorem 1.3) that if ∅ ̸ = π ̸ = P and r = r(π) = min P \ π, then r − 1 ⩽ BS(π) ⩽ max{11, 2(r − 2)}.…”

Section: § 1 Introductionmentioning

confidence: 99%

“…The lower bound for BS(π) in the last inequality can be obtained from the following assertion: 2 if r ⩾ 3, then any r − 2 transpositions in the symmetric group S r generate a π-subgroup, whereas O π (S r ) = 1 (see [32], Proposition 1.1). This estimate shows that the Baer-Suzuki width of the class of π-groups can be arbitrarily large for suitable π.…”

Section: § 1 Introductionmentioning

confidence: 99%

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On the sharp Baer-Suzuki theorem for the $\pi$-radical of a finite group

Yang

Ревин

et al. 2023

Sb. Math.

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Let $\pi$ be a proper subset of the set of prime numbers. Denote by $r$ the least prime not contained in $\pi$ and set $m=r$ for $r=2$ and $3$ and $m=r-1$ for $r\geqslant5$. The conjecture under consideration claims that a conjugacy class $D$ of a finite group $G$ generates a $\pi$-subgroup of $G$ (equivalently, is contained in the $\pi$-radical) if and only if any $m$ elements of $D$ generate a $\pi$-group. It is shown that this conjecture holds if every non-Abelian composition factor of $G$ is isomorphic to a sporadic, an alternating, a linear, or a unitary simple group. Bibliography: 49 titles.

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Section: § 1 Introductionmentioning

confidence: 99%

“…In [32], Theorem 1.2, the authors proved that for any π ⊆ P there always exists a nonnegative integer m = m(π) such that for an arbitrary group G O π (G) = {x ∈ G | ⟨x g1 , . .…”

Section: § 1 Introductionmentioning

confidence: 99%

Section: § 1 Introductionmentioning

confidence: 99%

Section: § 1 Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On the sharp Baer-Suzuki theorem for the $\pi$-radical of a finite group

Yang

Ревин

et al. 2023

Sb. Math.

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“…The main result of [10] constitutes finding explicit, albeit not always best possible, upper bounds on α S (x) for all nonabelian simple groups S. These bounds and their refinements have been extensively used in applications of the classification of finite simple groups. For example, they are substantially used in proofs of various analogues of the famous Baer-Suzuki theorem, see [3,5,6,7,8,9,17,18,19,16,15]. For practical use, the estimates on α S (x) from [10] are not always sufficient.…”

Section: Introductionmentioning

confidence: 99%

On generations by conjugate elements in almost simple groups with socle ²𝐹₄(𝑞²)′

Ревин

Zavarnitsine

2023

Journal of Group Theory

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We prove that if L = F 4 2 ⁢ ( 2 2 ⁢ n + 1 ) ′ L={}^{2}F_{4}(2^{2n+1})^{\prime} and 𝑥 is a nonidentity automorphism of 𝐿, then G = ⟨ L , x ⟩ G=\langle L,x\rangle has four elements conjugate to 𝑥 that generate 𝐺. This result is used to study the following conjecture about the 𝜋-radical of a finite group. Let 𝜋 be a proper subset of the set of all primes and let 𝑟 be the least prime not belonging to 𝜋. Set m = r m=r if r = 2 r=2 or 3 and m = r − 1 m=r-1 if r ⩾ 5 r\geqslant 5 . Supposedly, an element 𝑥 of a finite group 𝐺 is contained in the 𝜋-radical O π ⁡ ( G ) \operatorname{O}_{\pi}(G) if and only if every 𝑚 conjugates of 𝑥 generate a 𝜋-subgroup. Based on the results of this and previous papers, the conjecture is confirmed for all finite groups whose every nonabelian composition factor is isomorphic to a sporadic, alternating, linear, unitary simple group, or to one of the groups of type B 2 2 ⁢ ( 2 2 ⁢ n + 1 ) {}^{2}B_{2}(2^{2n+1}) , G 2 2 ⁢ ( 3 2 ⁢ n + 1 ) {}^{2}G_{2}(3^{2n+1}) , F 4 2 ⁢ ( 2 2 ⁢ n + 1 ) ′ {}^{2}F_{4}(2^{2n+1})^{\prime} , G 2 ⁢ ( q ) G_{2}(q) , or D 4 3 ⁢ ( q ) {}^{3}D_{4}(q) .

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Toward a Sharp Baer–Suzuki Theorem for the π-Radical: Exceptional Groups of Small Rank

Wang,

Guo,

Revin

2023

Algebra Logic

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Baer-Suzuki theorem for the π-radical

Cited by 9 publications

References 24 publications

On the sharp Baer-Suzuki theorem for the $\pi$-radical of a finite group

On the sharp Baer-Suzuki theorem for the $\pi$-radical of a finite group

On generations by conjugate elements in almost simple groups with socle ²𝐹₄(𝑞²)′

Toward a Sharp Baer–Suzuki Theorem for the π-Radical: Exceptional Groups of Small Rank

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Baer-Suzuki theorem for the π-radical

Cited by 9 publications

References 24 publications

On the sharp Baer-Suzuki theorem for the $\pi$-radical of a finite group

On the sharp Baer-Suzuki theorem for the $\pi$-radical of a finite group

On generations by conjugate elements in almost simple groups with socle 2𝐹4(𝑞2)′

Toward a Sharp Baer–Suzuki Theorem for the π-Radical: Exceptional Groups of Small Rank

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On generations by conjugate elements in almost simple groups with socle ²𝐹₄(𝑞²)′