On Solvability of Finite Groups with Few Non-Normal Subgroups

For a finite group G, let v(G) denote the number of conjugacy classes of non-normal subgroups of G and vc(G) denote the number of conjugacy classes of non-normal noncyclic subgroups of G. In this paper, we show that every finite group G satisfying v(G) ≤2|π(G)| or vc(G) ≤ |π(G)| is solvable, and for a finite nonsolvable group G, v(G) = 2|π(G)| +1 if and only if G ? A ₅.     

Frattini and related subgroups of mapping class groups

Let Γ_g,b denote the orientation-preserving mapping class group of a closed orientable surface of genus g with b punctures. For a group G let Φ_f(G) denote the intersection of all maximal subgroups of finite index in G. Motivated by a question of Ivanov as to whether Φ_f(G) is nilpotent when G is a finitely generated subgroup of Γ_g,b, in this paper we compute Φ_f(G) for certain subgroups of Γ_g,b. In particular, we answer Ivanov’s question in the affirmative for these subgroups of Γg,b.     

On finite groups with non-subnormal subgroups

For a finite group G, let n(G) denote the number of conjugacy classes of non-subnormal subgroups of G. In this paper, we show that a finite group G satisfying n(G)≤2|π(G)| is solvable, and for a finite non-solvable group G, n(G) = 2|π(G)|+1 if and only if G?A₅.     

On Solvable R* Groups of Finite Rank

Abstract

For an element a of a group G,let S(a) denote the semigroup generated by all conjugates of a in G. We prove that if G is solvable of finite rank and 1 ? S(a) for all 1 ≠ a ∈ G,then ?a ^G ?/?b ^G ? is a periodic group for every b ∈ S(a). Conversely if every two generator subgroup of a finitely generated torsion-free solvable group G has this property then G has finite rank,and if every finitely generated subgroup has this property then every partial order on G can be extended to a total order.     

On Two Theorems of Finite Solvable Groups

For a finite group G, let T（G） denote a set of primes such that a prime p belongs to T（G） if and only if p is a divisor of the index of some maximal subgroup of G. It is proved that if G satisfies any one of the following conditions： （1） G has a p-complement for each p∈T（G）; （2）│T（G）│= 2： （3） the normalizer of a Sylow p-subgroup of G has prime power index for each odd prime p∈T（G）; then G either is solvable or G/Sol（G）≌PSL（2, 7） where Sol（G） is the largest solvable normal subgroup of G.     

Characterization of A5 and SL(2,5) by the number of conjugacy classes of non-cyclic subgroups

For a finite group G, let ν_c(G) denote the number of conjugacy classes of non-normal non-cyclic subgroups of G. We show that for every finite non-solvable group G, ν_c(G) = |π(G)|+1 if and only if G?A₅, the alternating group on 5 letters, or SL(2,5).     

On a group with three supersolvable subgroups of pairwise relatively prime indices

In this paper, we show that if G is a finite group with three supersolvable subgroups of pairwise relatively prime indices in G and G′ is nilpotent, then G is supersolvable. Let π(G) denote the set of prime divisors of |G| and max(π(G)) denote the largest prime divisor of |G|. We also establish that if G is a finite group such that G has three supersolvable subgroups H, K, and L whose indices in G are pairwise relatively prime, q \nmid p-1{q \nmid p-1} where p =  max(π(G)) and q = max(π(L)) with L a Hall p′-subgroup of G, then G is supersolvable.     

Factorization of Periodic Locally Solvable Groups by Locally Nilpotent and Nilpotent Subgroups

We establish a series of new results concerning periodic locally solvable and finite solvable groups G = AB with locally nilpotent or nilpotent subgroups A and B.     

The number of conjugacy classes of non-normal subgroups in nilpotent groups

In a recent paper, Rolf Brandi classified all finite groups having exactly one conjugacy class of nonnormal subgroups, and conjectured thatfor a nilpotent group G of nilpotency class c = c(G) the number v(G) = vof conjugacy classes of nonnormal subgroups satisfies the inequality v(G) ≥ c(G) – 1 (with the exception of the Hamiltonian groups, of course). The purpose of this paper is to establish this conjecture and to decide when this inequality is sharp.     

Derived length and conjugacy class sizes

Let G be a finite solvable group, and let F(G) be its Fitting subgroup. We prove that there is a universal bound for the derived length of G/F(G) in terms of the number of distinct conjugacy class sizes of G. This result is asymptotically best possible. It is based on the following result on orbit sizes in finite linear group actions: If G is a finite solvable group and V a finite faithful irreducible G-module of characteristic r, then there is a universal logarithmic bound for the derived length of G in terms of the number of distinct r^′-parts of the orbit sizes of G on V. This is a refinement of the author's previous work on orbit sizes.     

Nilpotent Length of a Finite Solvable Group with a Frobenius Group of Automorphisms

We prove that a finite solvable group G admitting a Frobenius group FH of automorphisms of coprime order with kernel F and complement H such that [G, F] = G and C _{C G (F)}(h) = 1 for all nonidentity elements h ∈ H, is of nilpotent length equal to the nilpotent length of the subgroup of fixed points of H.     

The Conjugacy Class Number k(G): A Different Perspective

For a finite group G, let m(G) denote the least positive integer such that the union of any m(G) distinct nontrivial conjugacy classes of G together with the identity of G is a subgroup of G. We prove that m(G) = k(G) ?1 for all m(G) ≥2.     

Maximal covers of finite groups

Abstract

Let λ(G) be the maximum number of subgroups in an irredundant covering of the finite group G. We prove that if G is a group with λ(G) ≤ 6, then G is supersolvable. We also describe the structure of groups G with λ(G) = 6. Moreover, we show that if G is a group with λ(G)?<?31, then G is solvable.     

Free pro-p groups as galois groups over ℚ(p)(t)

Letp be a prime and let ℚ(p) denote the maximalp-extension of ℚ. We prove that for every primep, the free pro-p group on countably many generators is realizable as a regular extension of ℚ(p)(t). As a consequence, if ℚ _nil denotes the maximal nilpotent extension of ℚ, then every finite nilpotent group is realizable as a regular extension of ℚ _nil (t).     

Cycles in bipartite graphs

For k an integer, let G(a, b, k) denote a simple bipartite graph with bipartition (A, B) where |A| = a ≥ 2, |B| = b ≥ k ≥ 2, and each vertex of A has degree at least k. We prove two results concerning the existence of cycles in G(a, b, k).     

The average degree of an irreducible character of a finite group

Given a finite group G, we write acd(G) to denote the average of the degrees of the irreducible characters of G. We show that if acd(G) ≤ 3, then G is solvable. Also, if acd(G) < 3/2, then G is supersolvable, and if acd(G) < 4/3, then G is nilpotent.     

Correspondences Between 2-Brauer Characters of Solvable Groups

Let G be a finite solvable group and let p be a prime. Let P ∈ Syl _p (G) and N = N _G (P). We prove that there exists a natural bijection between the 2-Brauer irreducible characters of p′-degree of G and those of N _G (P).     

Notes on the minimal number of ramified primes in some l-extensions of Q

For a finite l-group G, let ram ^t (G) denote the minimal integer such that G can be realized as the Galois group of a tamely ramified extension of Q ramified only at ram ^t (G) finite primes. We study the upper bound of ram ^t (G) and give an improvement of the result of Plans. We also give the best bound of ram ^t (G) for all 3-groups G of order less than or equal to 3⁵. Received: 14 September 2007     

The structure of finite groups with trait of non-normal subgroups

In this paper, we investigate the finite groups all of whose non-normal nilpotent subgroups are cyclic. We show that such groups are solvable with cyclic centers. If G is a non-supersolvable group, then G has only one non-cyclic Sylow subgroup which is either isomorphic to Q₈ or is of type (q, q).