On the number of conjugates defining the solvable radical of a finite group (original) (raw)

A commutator description of the solvable radical of a finite group

2006

We are looking for the smallest integer k > 1 providing the following characterization of the solvable radical R(G) of any finite group G: R(G) coincides with the collection of g ∈ G such that for any k elements a1, a2,..., ak ∈ G the subgroup generated by the elements g, aiga −1 i, i = 1,...,k, is solvable. We consider a similar problem of finding the smallest integer ℓ> 1 with the property that R(G) coincides with the collection of g ∈ G such that for any ℓ elements b1, b2,..., bℓ ∈ G the subgroup generated by the commutators [g, bi], i = 1,..., ℓ, is solvable. Conjecturally, k = ℓ = 3. We prove that both k and ℓ are at most 7. In particular, this means that a finite group G is solvable if and only if in each conjugacy class of G every 8 elements generate a

Characterizations of the solvable radical

Proceedings of the American Mathematical Society, 2010

We prove that there exists a constant k k with the property: if C \mathcal {C} is a conjugacy class of a finite group G G such that every k k elements of C \mathcal {C} generate a solvable subgroup, then C \mathcal {C} generates a solvable subgroup. In particular, using the Classification of Finite Simple Groups, we show that we can take k = 4 k=4 . We also present proofs that do not use the Classification Theorem. The most direct proof gives a value of k = 10 k=10 . By lengthening one of our arguments slightly, we obtain a value of k = 7 k=7 .

A description of Baer–Suzuki type of the solvable radical of a finite group

Journal of Pure and Applied Algebra, 2009

We obtain the following characterization of the solvable radical R(G) of any finite group G: R(G) coincides with the collection of all g ∈ G such that for any 3 elements a 1 , a 2 , a 3 ∈ G the subgroup generated by the elements g, a i ga −1 i , i = 1, 2, 3, is solvable. In particular, this means that a finite group G is solvable if and only if in each conjugacy class of G every 4 elements generate a solvable subgroup. The latter result also follows from a theorem of P. Flavell on {2, 3}-elements in the solvable radical of a finite group (which does not use the classification of finite simple groups).

From Thompson to Baer-Suzuki: a sharp characterization of the solvable radical

arXiv: Group Theory, 2009

We prove that an element g of prime order > 3 belongs to the solvable radical R(G) of a finite (or, more generally, a linear) group if and only if for every x ∈ G the subgroup generated by g, xgx −1 is solvable. This theorem implies that a finite (or a linear) group G is solvable if and only if in each conjugacy class of G every two elements generate a solvable subgroup.

Characterization of Solvable Groups and Solvable Radical

International Journal of Algebra and Computation, 2013

We give a survey of new characterizations of finite solvable groups and the solvable radical of an arbitrary finite group which were obtained over the past decade. We also discuss generalizations of these results to some classes of infinite groups and their analogues for Lie algebras. Some open problems are discussed as well.

New Trends in Characterization of Solvable Groups

webdoc.sub.gwdg.de

Abstract. We give a survey of new characterizations of finite solvable groups and the solvable radical of an arbitrary finite group which were obtained over the past decade. We also discuss generalizations of these results to some classes of infinite groups and their analogues for ...

Thompson-like characterizations of the solvable radical

Journal of Algebra, 2006

We prove that the solvable radical of a finite group G coincides with the set of elements y having the following property: for any x ∈ G the subgroup of G generated by x and y is solvable. This confirms a conjecture of Flavell. We present analogues of this result for finite-dimensional Lie algebras and some classes of infinite groups. We also consider a similar problem for pairs of elements.

Thompson-like characterization of the solvable radical

Journal of Algebra - J ALGEBRA, 2005

We prove that the solvable radical of a finite group G coincides with the set of elements y having the following property: for any x in G the subgroup of G generated by x and y is solvable. We present analogues of this result for finite dimensional Lie algebras and some classes of infinite groups. We also consider a similar problem for pairs of elements.

Two problems on finite groups with k conjugate classes

Journal of the Australian Mathematical Society, 1968

Let G be a finite group of order g having exactly k conjugate classes. Let π(G) denote the set of prime divisors of g. K. A. Hirsch [4] has shown that By the same methods we prove g ≡ k modulo G.C.D. {(p–1)2 p ∈ π(G)} and that if G is a p-group, g = h modulo (p−1)(p2−1). It follows that k has the form (n+r(p−1)) (p2−1)+pe where r and n are integers ≧ 0, p is a prime, e is 0 or 1, and g = p2n+e. This has been established using representation theory by Philip Hall [3] (see also [5]). If then simple examples show (for 6 ∤ g obviously) that g ≡ k modulo σ or even σ/2 is not generally true.