<Emphasis Type="Italic">θ</Emphasis>-Pairs for 2-maximal subgroups of finite groups

Let H, A and B be subgroups of a group G. We call the pair (A, B) a θ-pair for H in G if: (i) \({\langle H, A\rangle=G}\) and B = (A ∩ H) _G ; (ii) if A ₁/B is a proper subgroup of A/B and \({{A_1/B \vartriangleleft G/B}}\), then \({G\neq \langle H, A_1\rangle}\). In this paper, we study the θ-pairs for 2-maximal subgroups of a group, which imply a group to be solvable or supersolvable.     

A note on commuting automorphisms of some finite <Emphasis Type="Italic">p</Emphasis>-groups

An automorphism α of a group G is called a commuting automorphism if each element x in G commutes with its image α(x) under α. Let A(G) denote the set of all commuting automorphisms of G. Rai [Proc. Japan Acad., Ser. A 91 (5), 57–60 (2015)] has given some sufficient conditions on a finite p-group G such that A(G) is a subgroup of Aut(G) and, as a consequence, has proved that, in a finite p-group G of co-class 2, where p is an odd prime, A(G) is a subgroup of Aut(G). We give here very elementary and short proofs of main results of Rai.     

On color-preserving automorphisms of Cayley graphs of odd square-free order

An automorphism \(\alpha \) of a Cayley graph \(\mathrm{Cay}(G,S)\) of a group G with connection set S is color-preserving if \(\alpha (g,gs) = (h,hs)\) or \((h,hs^{-1})\) for every edge \((g,gs)\in E(\mathrm{Cay}(G,S))\). If every color-preserving automorphism of \(\mathrm{Cay}(G,S)\) is also affine, then \(\mathrm{Cay}(G,S)\) is a Cayley color automorphism (CCA) graph. If every Cayley graph \(\mathrm{Cay}(G,S)\) is a CCA graph, then G is a CCA group. Hujdurovi? et al. have shown that every non-CCA group G contains a section isomorphic to the non-abelian group \(F_{21}\) of order 21. We first show that there is a unique non-CCA Cayley graph \(\Gamma \) of \(F_{21}\). We then show that if \(\mathrm{Cay}(G,S)\) is a non-CCA graph of a group G of odd square-free order, then \(G = H\times F_{21}\) for some CCA group H, and \(\mathrm{Cay}(G,S) = \mathrm{Cay}(H,T)\mathbin {\square }\Gamma \).     

Note: A conjecture on partitions of groups

We conjecture that every infinite group G can be partitioned into countably many cells \(G = \bigcup\limits_{n \in \omega } {A_n }\) such that cov(A _n A _n ^?1 ) = |G| for each n ∈ ω Here cov(A) = min{|X|: X} ? G, G = X A}. We confirm this conjecture for each group of regular cardinality and for some groups (in particular, Abelian) of an arbitrary cardinality.     

Finite <Emphasis Type="Italic">p</Emphasis>-groups all of whose maximal abelian subgroups are soft

A subgroup A of a p-group G is said to be soft in G if C _G (A) = A and |N _G (A/A| = p. In this paper we determined finite p-groups all of whose maximal abelian subgroups are soft; see Theorem A and Proposition 2.4.     

On an open problem of Guo-Skiba

Let G be a finite group, and let A be a proper subgroup of G. Then any chief factor H/A _G of G is called a G-boundary factor of A. For any Gboundary factor H/A _G of A, the subgroup (A ∩ H)/A _G of G/ A _G is called a G-trace of A. In this paper, we prove that G is p-soluble if and only if every maximal chain of G of length 2 contains a proper subgroup M of G such that either some G-trace of M is subnormal or every G-boundary factor of M is a p′-group. This result give a positive answer to a recent open problem of Guo and Skiba. We also give some new characterizations of p-hypercyclically embedded subgroups.     

New characterization of <Emphasis Type="Italic">S</Emphasis><Subscript>4</Subscript>(<Emphasis Type="Italic">q</Emphasis>) by its noncommuting graph

Let G be a nonabelian group, and associate the noncommuting graph ?(G) with G as follows: the vertex set of ?(G) is G\Z(G) with two vertices x and y joined by an edge whenever the commutator of x and y is not the identity. Let S ₄(q) be the projective symplectic simple group, where q is a prime power. We prove that if G is a group with ?(G) ? ?(S ₄(q)) then G ? S ₄(q).     

2-Recognizability by prime graph of <Emphasis Type="Italic">PSL</Emphasis>(2, <Emphasis Type="Italic">p</Emphasis><Superscript>2</Superscript>)

Let G be a finite group and let Γ(G) be the prime graph of G. Assume p prime. We determine the finite groups G such that Γ(G) = Γ(PSL(2, p ²)) and prove that if p ≠ 2, 3, 7 is a prime then k(Γ(PSL(2, p ²))) = 2. We infer that if G is a finite group satisfying |G| = |PSL(2, p ²)| and Γ(G) = Γ(PSL(2, p ²)) then G ? PSL(2, p ²). This enables us to give new proofs for some theorems; e.g., a conjecture of W. Shi and J. Bi. Some applications are also considered of this result to the problem of recognition of finite groups by element orders.     

Products of F*(<Emphasis Type="Italic">G</Emphasis>)-subnormal subgroups of finite groups

A subgroup H of a finite group G is called F*(G)-subnormal if H is subnormal in HF*(G). We show that if a group Gis a product of two F*(G)-subnormal quasinilpotent subgroups, then G is quasinilpotent. We also study groups G = AB, where A is a nilpotent F*(G)-subnormal subgroup and B is a F*(G)-subnormal supersoluble subgroup. Particularly, we show that such groups G are soluble.     

Quasirecognition by prime graph of <Emphasis Type="Italic">L</Emphasis><Subscript>10</Subscript>(2)

Let G be a finite group. The prime graph of G is denoted by Γ(G). The main result we prove is as follows: If G is a finite group such that Γ(G) = Γ(L ₁₀(2)) then G/O ₂(G) is isomorphic to L ₁₀(2). In fact we obtain the first example of a finite group with the connected prime graph which is quasirecognizable by its prime graph. As a consequence of this result we can give a new proof for the fact that the simple group L ₁₀(2) is uniquely determined by the set of its element orders.     

On Degrees of Growth of Finitely Generated Groups

We prove that for an arbitrary function ρ of subexponential growth there exists a group G of intermediate growth whose growth function satisfies the inequality v _G,S (n) ? ρ(n) for all n. For every prime p, one can take G to be a p-group; one can also take a torsion-free group G. We also discuss some generalizations of this assertion.     

Groups equal to a product of three conjugate subgroups

Let G be a finite non-solvable group. We prove that there exists a proper subgroup A of G such that G is the product of three conjugates of A, thus replacing an earlier upper bound of 36 with the smallest possible value. The proof relies on an equivalent formulation in terms of double cosets, and uses the following theorem which is of independent interest and wider scope: Any group G with a BN-pair and a finite Weyl group W satisfies \(G = {\left( {B{n_0}B} \right)^2} = B{B^{{n_0}}}B\) where n₀ is any preimage of the longest element of W. The proof of the last theorem is formulated in the dioid consisting of all unions of double cosets of B in G. Other results on minimal length product covers of a group by conjugates of a proper subgroup are given.     

On the $${{\varvec{}}}$$-length of the mutually ermutable roduct of two $${{\varvec{}}}$$-soluble grous

Let a finite group \(G=AB\) be the mutually permutable product of two p-soluble subgroups A and B for some prime p. We give a bound of the p-length of G from the p-lengths of A and B.     

*-regularity of certain generalized group algebras

Let B be a *-semisimple Banach algebra with a bounded approximate identity and \({\alpha: G \longrightarrow {\rm Aut}_{*}(B)}\) (isometric *-automorphisms group of B) an action of a locally group G on B. Let (D, G, γ) be the associated dynamical system, where D = C ₀(G, B) is the Banach *-algebra of all continuous B-valued functions on G vanishing at infinity and the action γ : G → Aut D is given by γ _s (y)(t) = α _s (y(s ^?1 t)) for \({y \in D}\) and \({s, t \in G}\) . Recall that B is said to be *-regular if the natural mapping \({I\in {\rm Prim} \, C^{*}(B) \mapsto I\cap B\in {\rm Prim}_{*}(B)}\) is a homeomorphism under the hull-kernel topology. When G is amenable, we show that if B is *-regular, then the generalized group algebra L ¹(G, D; γ) is *-regular. The converse is also true if we further assume that G is countable discrete. Finally the case of compact groups is studied.     

Thompson’s conjecture for the alternating group of degree 2<Emphasis Type="Italic">p</Emphasis> and 2<Emphasis Type="Italic">p</Emphasis>+1

For a finite group G denote by N(G) the set of conjugacy class sizes of G. In 1980s, J.G.Thompson posed the following conjecture: If L is a finite nonabelian simple group, G is a finite group with trivial center and N(G) = N(L), then G ? L. We prove this conjecture for an infinite class of simple groups. Let p be an odd prime. We show that every finite group G with the property Z(G) = 1 and N(G) = N(A _i ) is necessarily isomorphic to A _i , where i ∈ {2p, 2p + 1}.     

Elliptic function of level 4

The article is devoted to the theory of elliptic functions of level n. An elliptic function of level n determines a Hirzebruch genus called an elliptic genus of level n. Elliptic functions of level n are also of interest because they are solutions of the Hirzebruch functional equations. The elliptic function of level 2 is the Jacobi elliptic sine function, which determines the famous Ochanine–Witten genus. It is the exponential of the universal formal group of the form F(u, v) = (u² ? v²)/(uB(v) ? vB(u)), B(0) = 1. The elliptic function of level 3 is the exponential of the universal formal group of the form F(u, v) = (u²A(v) ? v²A(u))/(uA(v)² ? vA(u)²), A(0) = 1, A″(0) = 0. In the present study we show that the elliptic function of level 4 is the exponential of the universal formal group of the form F(u, v) = (u²A(v) ? v²A(u))/(uB(v) ? vB(u)), where A(0) = B(0) = 1 and for B′(0) = A″(0) = 0, A′(0) = A₁, and B″(0) = 2B₂ the following relation holds: (2B(u) + 3A₁u)² = 4A(u)³ ? (3A₁² ? 8B₂)u²A(u)². To prove this result, we express the elliptic function of level 4 in terms of the Weierstrass elliptic functions.     

On the length of a finite group and of its 2-generator subgroups

The nonsoluble length λ(G) of a finite group G is defined as the minimum number of nonsoluble factors in a normal series of G each of whose quotients either is soluble or is a direct product of nonabelian simple groups. The generalized Fitting height of a finite group G is the least number h = h* (G) such that F* _h (G) = G, where F* ₁ (G) = F* (G) is the generalized Fitting subgroup, and F* _i+1(G) is the inverse image of F* (G/F*_i (G)). In the present paper we prove that if λ(J) ≤ k for every 2-generator subgroup J of G, then λ(G) ≤ k. It is conjectured that if h* (J) ≤ k for every 2-generator subgroup J, then h* (G) ≤ k. We prove that if h* (〈x, x^g 〉) ≤ k for allx, g ∈ G such that 〈x, x^g 〉 is soluble, then h* (G) is k-bounded.     

On the number of irreducible characters in a 3-block with a minimal nonabelian defect group

Let B be a 3-block of a finite group G with a defect group D. In this paper, we are mainly concerned with the number of characters in a particular block, so we shall use Isaacs' approach to block structure. We consider the block B of a group G as a union of two sets, namely a set of irreducible ordinary characters of G having cardinality k(B) and a set of irreducible Brauer characters of G having cardinality l(B). We calculate k(B) and l(B) provided that D is normal in G and D■x, y, z|x~(3n)=y~(3m)= z~3= [x, z] = [y, z] = 1, [x, y] = z(n m ≥ 2).     

The Automorphism Group of a Class of Nilpotent Groups with Infinite Cyclic Derived Subgroups

The automorphism group of a class of nilpotent groups with infinite cyclic derived subgroups is determined. Let G be the direct product of a generalized extraspecial Z-group E and a free abelian group A with rank m, where E ={(1 kα_1 kα_2 ··· kα_nα_(n+1) 0 1 0 ··· 0 α_(n+2)...............000...1 α_(2n+1)000...01|αi∈ Z, i = 1, 2,..., 2 n + 1},where k is a positive integer. Let AutG G be the normal subgroup of Aut G consisting of all elements of Aut G which act trivially on the derived subgroup G of G, and AutG/ζ G,ζ GG be the normal subgroup of Aut G consisting of all central automorphisms of G which also act trivially on the center ζ G of G. Then(i) The extension 1→ Aut_(G') G→ AutG→ Aut(G')→ 1 is split.(ii) Aut_(G') G/Aut_(G/ζ G,ζ G)G≌Sp(2 n, Z) ×(GL(m, Z)■(Z~)m).(iii) Aut_(G/ζ G,ζ GG/Inn G)≌(Z_k)~(2n)⊕(Z)~(2nm).     

X-quasinormal subgroups

Considering two subgroups A and B of a group G and ? ≠ X ? G, we say that A is X-permutable with B if AB ^x = B ^x A for some element x ∈ X. We use this concept to give new characterizations of the classes of solvable, supersolvable, and nilpotent finite groups.