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Let be the prime graph associated with a finite group and be the degree pattern of . A finite group is said to be -fold -characterizable if there exist exactly nonisomorphic groups such that and . The purpose of this paper is two-fold. First, it shows that the symmetric group is -fold -charaterizable. Second, it shows that there exist many infinite families of alternating and symmetric groups, and , which are -fold -characterizable with . [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of finite wave packet groups for positive prime integers, as groups of cyclic dilations, translations and modulations. We will present a unified group theoretical approach for the cyclic wave packet transform on finite Abelian groups of prime order. Then, we study basic properties of this windowed transform. [ABSTRACT FROM AUTHOR]

Let G be a finite group and πe(G) be the set of element orders of G. Assume that k ∈ πe(G) and mk(G) is the number of elements of order k in G. Set (G) ≔ {mk(G) | k ∈ πe(G)}, we call (G) the set of numbers of elements with same order. In this paper, we give a new characterization of simple linear group L2(2a) by its order |L2(2a)| and the set (L2(2a)), where either 2a - 1 or 2a + 1 is a prime. [ABSTRACT FROM AUTHOR]

For a finite group G, we denote by p(G) the minimal degree of faithful permutation representations of G, and denote by q(G) and c(G), the minimal degree of faithful representation of G by quasi-permutation matrices over the rational field ℚ and the complex field ℂ, respectively. In this paper, we calculate p(G), q(G) and c(G) for the groups of order p5, where p is an odd prime. [ABSTRACT FROM AUTHOR]

In this paper, we generalize the main results of Guo and Isaacs, Archiv der Mathematik 105 (2015) 215-222. We also obtain some criteria for -nilpotency of a finite group. [ABSTRACT FROM AUTHOR]

In this paper, finite p-groups in which any two noncommutative elements generate an inner abelian group of order p4 are classified. [ABSTRACT FROM AUTHOR]

In this paper, a characteristic subgroup of a finite group G is defined, which is the intersection of the normalizers of the nilpotent residuals of all subgroups of G, and the properties of and the relationship between and the group G are investigated. [ABSTRACT FROM AUTHOR]

In this paper, we prove a conjecture of Thompson for an infinite class of simple groups of Lie type. In fact, we show that every finite group G with the property Φ(G) ∩ Z(G) = 1 and N(G) = N(Dn(q)) is isomorphic to Z(G) × Dn(q), where n ≥ 5 and n ≠ 8. Note that N(G), Φ(G) and Z(G) are the set of lengths of conjugacy classes of G, Frattini subgroup of G and the center of G, respectively. [ABSTRACT FROM AUTHOR]

A translation in an algebraic signature is a finite conjunction of equations in one variable. On a quasivariety K, a translation τ naturally induces a deductive system, called the τ-assertional logic of K. Two quasivarieties are τ-assertionally equivalent if they have the same τ-assertional logic. This paper is a study of assertional equivalence. It characterizes the quasivarieties equivalent to ones with various desirable properties, such as τ-regularity (a general form of point regularity). Special attention is paid to structural properties of quasivarieties that are assertionally equivalent to their varietal closures under an indicated translation. [ABSTRACT FROM AUTHOR]

Using the technique of linear limits of characters due to Dade and Loukaki, we give some conditions on the supersolvable residual of a finite solvable group G that is sufficient to guarantee that G is an M-group. The monomiality of normal subgroups and Hall subgroups of the group G are also determined. [ABSTRACT FROM AUTHOR]

In this paper, finite p-groups in which any two noncommutative elements generate an inner abelian group of order p4 are classified. [ABSTRACT FROM AUTHOR]

The concept of configuration was first introduced by Rosenblatt and Willis to give a characterization for the amenability of groups. Then Rejali and Yousofzadeh introduced the notion of two-sided configuration to study the normal subsets of a group. In [A, Abdollahi, A. Rejali and G. A. Willis, Group properties characterised by configurations, Illinois J. Math.48(3) (2004) 861–873.], the authors have asked that if two configuration equivalent groups are isomorphic? We show that if G and H have same two-sided configuration sets and N is a normal subgroup of G with polycyclic or FC quotient, then there is a normal subgroup 𝔑 of H such that G / N ≅ H / 𝔑. Also, we show that if G and H are two-sided equivalent groups, and if one of them is polycyclic or FC, then they are isomorphic. [ABSTRACT FROM AUTHOR]

Let be a finite group, whose order has prime divisors. In this paper, we prove that if has a -complement for prime divisors of and has no section isomorphic to . Then is solvable, which generalizes a theorem of Hall. [ABSTRACT FROM AUTHOR]

Let F, N, A and N2 denote the properties of being finite, nilpotent, abelian and nilpotent of classes at most 2, respectively. Firstly we consider the class of finitely generated FN-groups. We show that the property FC is closed under finite extensions, and extend this result to finitely generated NF-groups. Secondly we prove that a finitely generated NF-group G is in the class ((FC)F, ∞) if and only if G is an FA-group. Finally we prove that a finitely generated NF-group in the class ((FC)F, ∞)* is an FN2-group. Moreover, G/Z2(G) is finite. [ABSTRACT FROM AUTHOR]

The prime graph question asks whether the Gruenberg-Kegel graph of an integral group ring , i.e. the prime graph of the normalized unit group of , coincides with that one of the group . In this note, we prove for finite groups a reduction of the prime graph question to almost simple groups. We apply this reduction to finite groups whose order is divisible by at most three primes and show that the Gruenberg-Kegel graph of such groups coincides with the prime graph of . [ABSTRACT FROM AUTHOR]

Motivated by Isaacs and Passman's characterization of finite groups all of whose nonlinear complex irreducible characters have prime degrees, we investigate finite groups with exactly one character degree that is not a prime. We show that either is solvable with or for distinct primes , or up to an abelian direct factor, is isomorphic to the alternating group . [ABSTRACT FROM AUTHOR]

We describe the structure of finite groups in which all normal closures of non-normal subgroups have the same order. In particular, in the case of -groups, we give a complete classification. [ABSTRACT FROM AUTHOR]

Set-theoretic solutions of the quantum dynamical Yang-Baxter equation are analyzed by means of dynamical analogues of braces (=bijective 1-cocycles). It is shown that every abelian group acts on a universal parameter set such that the orbits correspond to the faithful irreducible dynamical braces with underlying abelian group . Braces correspond to the fixed points of this action. Relations between linear extendability of solutions, non-degeneracy, and existence of a double, are established for the dynamical case. [ABSTRACT FROM AUTHOR]

We study finite groups with given indices of 2-maximal subgroups. Suppose that every 2-maximal subgroup of a group has the property: if , then divides the square of a prime. Then is solvable. [ABSTRACT FROM AUTHOR]

Let a finite group act coprimely on a finite group . The Glauberman-Isaacs correspondence is a bijection from the set of -invariant irreducible characters of onto the set of irreducible characters of the centralizer of in . Let be a subgroup of . Composing from left to right, it follows that is an injection from into . We show that, in some cases, the map can be defined via the actions of some subgroups of containing on the centralizers in of some other such subgroups. We also show in many instances, such as odd or supersolvable and solvable, that this map is independent of the overgroup . [ABSTRACT FROM AUTHOR]

We prove that the -vector space generated by the multiple zeta values is generated by the finite real multiple zeta values introduced by Kaneko and Zagier. [ABSTRACT FROM AUTHOR]

The isomorphism types of all minimal nonabelian subgroups (-subgroups) of -groups are described. This allows us to classify the nonabelian p-groups, p > 2, that have no p isomorphic -subgroups of minimal order. In particular, if a p-group G is neither abelian nor -group and all its -subgroups are pairwise non-isomorphic, then either G ≅ 16, the semidihedral group of order 16, or G is a metacyclic 2-group of order ≥ 25. We also show that if a p-group G is neither abelian nor minimal nonabelian, then G is metacyclic if and only if all its -subgroups are metacyclic. [ABSTRACT FROM AUTHOR]

The concept of configuration was first introduced by Rosenblatt and Willis to give a condition for amenability of groups. We show that if G1 and G2 have the same configuration sets and H1 is a normal subgroup of G1 with abelian quotient, then there is a normal subgroup H2 of G2 such that $\frac{G_1}{H_1} \cong \frac{G_2}{H_2}$. Also configuration of FC-groups and isomorphism is studied. [ABSTRACT FROM AUTHOR]