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

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 a₁,a₂,a₃∈G the subgroup generated by the elements , 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).     

Group Rings with Solvable Unit Groups of Minimal Derived Length

Let F be a field of characteristic p > 2 and G a nonabelian nilpotent group containing elements of order p. Write F G for the group ring. The conditions under which the unit group ??(F G) is solvable are known, but only a few results have been proved concerning its derived length. It has been established that if G is torsion, the minimum derived length is ?log₂(p + 1)?, and this minimum occurs if and only if |G′| = p. In the present note, we show that the same holds if G has elements of infinite order.     

On the number of maximal subgroups of a finite solvable group

For a finite solvable group G and prime number p, we use elementary methods to obtain an upper bound for \mathfrak m_p(G){\mathfrak {m}_{p}(G)} , defined as the number of maximal subgroups of G whose index in G is a power of p. From this we derive an upper bound on the total number of maximal subgroups of a finite solvable group in terms of its order. This bound improves existing bounds, and we identify conditions on the order of a finite solvable group under which this bound is best possible.     

Groups whose elements are not conjugate to their powers

We call a finite group irrational if none of its elements is conjugate to a distinct power of itself. We prove that those groups are solvable and describe certain classes of these groups, where the above property is only required for p-elements, for p from a prescribed set of primes.     

Conjugacy class sizes and solvability of finite groups

Let G be a finite group and let G* be the set of elements of primary, biprimary and triprimary orders of G. We show that suppose that the conjugacy class sizes of G* are exactly {1, p ^a , n, p ^a n} with (p, n)?=?1 and a??? 0, then G is solvable.     

Nilpotent and solvable radicals in locally finite congruence modular varieties

The Loewy rank of a modular latticeL of finite height is defined as the leastn for which there exista ₀=0t, < ... r=1 inL such that each interval I[a_i, a_i+1] is a complemented lattice. In this paper, a generalized notion of Loewy rank is applied to obtain new results in the commutator theory of locally finite congruence modular varieties. LetV be a finitely generated congruence modular variety. We prove that every algebra inV has a largest nilpotent congruence and a largest solvable congruence. Moreover, there exist first order formulas which define these special congruences in every algebra ofV.     

Odd character correspondences in solvable groups

Let G be a finite solvable group, p be some prime, let P be a Sylow p-subgroup of G, and let N be its normalizer in G. Assume that N has odd order. Then, we prove that there exists a bijection from the set of all irreducible characters of G of degree prime to p to the set of all the irreducible characters of degree prime to p of N such that it preserves ± the degree modulo p, the field of values, and the Schur index over every field of characteristic zero. This strengthens a more general recent result [A. Turull, Character correspondences in solvable groups, J. Algebra 295 (2006) 157–178], but only for the case under consideration here. In addition, we prove some other strong character correspondences that have very good rationality properties. As one consequence, we prove that a solvable group G has a non-trivial rational irreducible character with degree prime to p if and only if the order of the normalizer of a Sylow p-subgroup of G has even order.     

On Thompson's Theorem

Let p be a prime divisor of the order of a finite group G. Thompson (1970, J. Algebra14, 129–134) has proved the following remarkable result: a finite group G is p-nilpotent if the degrees of all its nonlinear irreducible characters are divisible by p (in fact, in that case G is solvable). In this note, we prove that a group G, having only one nonlinear irreducible character of p′-degree is a cyclic extension of Thompson's group. This result is a consequence of the following theorem: A nonabelian simple group possesses two nonlinear irreducible characters χ₁ and χ₂ of distinct degrees such that p does not divide χ₁(1)χ₂(1) (here p is arbitrary but fixed). Our proof depends on the classification of finite simple groups. Some properties of solvable groups possessing exactly two nonlinear irreducible characters of p′-degree are proved. Some open questions are posed.     

Groups where each element is conjugate to its certain power

This paper deals with a rationality condition for groups. Let n be a fixed positive integer. Suppose every element g of the finite solvable group is conjugate to its nth power g ⁿ . Let p be a prime divisor of the order of the group. We conclude that the multiplicative order of n modulo p is small, or p is small.     

A characterization of the linear groups L 2(p)

Let G be a finite group and π _e (G) be the set of element orders of G. Let k ∈ π _e (G) and m _k be the number of elements of order k in G. Set nse(G):= {m _k : k ∈ π _e (G)}. In fact nse(G) is the set of sizes of elements with the same order in G. In this paper, by nse(G) and order, we give a new characterization of finite projective special linear groups L ₂(p) over a field with p elements, where p is prime. We prove the following theorem: If G is a group such that |G| = |L ₂(p)| and nse(G) consists of 1, p ² ? 1, p(p + ?)/2 and some numbers divisible by 2p, where p is a prime greater than 3 with p ≡ 1 modulo 4, then G ? L ₂(p).