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AbstractA subgroup X of a group G is called pronormal if X and Xg are conjugate in 〈X,Xg〉 for each element g of G. Pronormal subgroups play a relevant role in many problems of group theory; for instance, they arise naturally in the study of groups in which normality is a transitive relation, because a subgroup is normal if and only if it is subnormal and pronormal. Groups with only pronormal subgroups have been described by Kuzennyĭ and Subbotin (at least in the locally graded case) and the aim of this paper is to investigate the structure of uncountable locally graded groups of cardinality ℵ in which all subgroups of cardinality ℵ are pronormal. [ABSTRACT FROM AUTHOR]

AbstractLeibniz algebras are certain generalizations of Lie algebras. Motivated by the concept of subinvariance and ascendancy in group theory, Schenkman studied properties of subinvariant subalgebras of a Lie algebra. Kawamoto studied ascendency in Lie algebras. In this paper we define subinvariance and asendency in Leibniz algebras and study their properties. It is shown that the signature results on subinvariance and ascendency in Lie algebras have analogs for Leibniz algebras. [ABSTRACT FROM AUTHOR]

AbstractWe define, and denote by anz(G), the average number of zeros of characters of a finite group G as the number of zeros in the character table of G dividing the number of irreducible characters of G. In this paper, we classify odd-order nonabelian groups G with anz(G)≤2. [ABSTRACT FROM AUTHOR]

AbstractThe conjugacy relation is an important tool in the study of group theory and now has been generalized to semigroups in several methods. In particular, the p-conjugacy, o-conjugacy, c-conjugacy, i-conjugacy, n-conjugacy and r-conjugacy in semigroups are introduced and investigated extensively and some problems related to these conjugacy relations are proposed. The aim of this paper is to continue the study of conjugacy relations in semigroups around some of these problems. We first point out that p-conjugacy in a semigroup that is embeddable in a group, is not necessarily transitive. Then we obtain a simple sufficient condition under which the o-conjugacy is the universal relation and prove that this condition is also necessary in some natural classes of semigroups. Finally, we explore some properties of r-conjugacy in restriction semigroups, which extends some results on i-conjugacy in inverse semigroups. Our results have answered or partially answered some problems raised by Araújo, Kinyon, Konieczny and Malheiro in [Proc.Roy.Soc.Edinburgh Sect.A 147, 1169–1214, 2017] and [J.Algebra 533, 142–173, 2019]. [ABSTRACT FROM AUTHOR]

Given a construction f on groups, we say that a group G is f -realisable if there is a group H such that G ≅ f (H) , and completely f-realisable if there is a group H such that G ≅ f (H) and every subgroup of G is isomorphic to f (H 1) for some subgroup H 1 of H and vice versa. Denote by L (G) the absolute center of a group G, that is the set of elements of G fixed by all automorphisms of G. By using the structure of the automorphism group of a ZM-group, in this paper we prove that cyclic groups C N , N ∈ N * , are completely L-realisable. [ABSTRACT FROM AUTHOR]

Let G be a classical group of dimension d and let a = (a 1 , ... , a d) be differential indeterminates over a differential field F of characteristic zero with algebraically closed field of constants C. Further let A (a) be a generic element in the Lie algebra g (F 〈 a 〉) of G obtained from parametrizing a basis of g with the indeterminates a. It is known (cf. [5]) that the differential Galois group of y ′ = A (a) y over F 〈 a 〉 is G (C) . In this paper we construct a differential field extension L of F 〈 a 〉 such that the field of constants of L is C, the differential Galois group of y ′ = A (a) y over L is still the full group G (C) and A (a) is gauge equivalent over L to a matrix in normal form which we introduced in [13]. We also consider specializations of the coefficients of A (a) . [ABSTRACT FROM AUTHOR]

In this paper, all supercharacter theories of dicyclic groups T 4 n are described. These supercharacter theories are constructed by using the supercharacter theories of dihedral groups of order 4n which are classified by Lamar. [ABSTRACT FROM AUTHOR]

This paper is a new contribution to the partial Galois theory of groups. First, given a unital partial action αG of a finite group G on an algebra S such that S is an α G -partial Galois extension of S α G and a normal subgroup H of G, we prove that α G induces a unital partial action α G / H of G/H on the subalgebra of invariants S α H of S such that S α H is an α G / H -partial Galois extension of S α G . Second, assuming that G is abelian, we construct a commutative inverse semigroup T par (G , R) , whose elements are equivalence classes of α G -partial abelian extensions of a commutative algebra R. We also prove that there exists a group isomorphism between T par (G , R) / ρ and T(G, A), where ρ is a congruence on T par (G , R) and T(G, A) is the classical Harrison group of the G-isomorphism classes of the abelian extensions of a commutative algebra A. It is shown that the study of T par (G , R) reduces to the case where G is cyclic. The set of idempotents of T par (G , R) is also investigated. [ABSTRACT FROM AUTHOR]

We consider three families of groups: the Bianchi groups S L (2 , O) where O is the ring of integers of an imaginary, quadratic field; the groups S L ‡ (2 , O) where O is a ‡ -order of a definite, rational quaternion algebra with an orthogonal involution; and the groups S L (2 , O) where O is an order of a definite, rational quaternion algebra. We show that such groups are generated by elementary matrices if and only if O is semi-Euclidean (or ‡ -semi-Euclidean), which is a generalization of the usual notion of a Euclidean ring. The proofs are surprisingly simple and proceed by considering fundamental domains of Kleinian groups. [ABSTRACT FROM AUTHOR]

Let G = C p n be a finite cyclic p-group, and let Hol (G) denote its holomorph. In this work, we find and characterize the regular subgroups of Hol (G) that mutually normalize each other in the symmetric group Sym (G) . We represent regular subgroups of Hol (G) as vertices of a graph, and we connect a pair of them by an edge when they mutually normalize each other. The approach to construct this local normalizing graph relies on the theory of gamma functions, and the final result will contain all the information about the regular subgroups of Hol (G) in a compact form. Communicated by Alberto Facchini [ABSTRACT FROM AUTHOR]

The theory of Frobenius groups with Frobenius complements of even order largely reduces to tractable algebraic number theory. If we consider only Frobenius complements with an upper bound s on the number of distinct primes dividing the order of their commutator subgroups, then the proportion of these with odd order is less than 1/2s . A positive lower bound is also given. [ABSTRACT FROM AUTHOR]

Factor systems are a tool for working on the extension problem of algebraic structures such as groups, Lie algebras, and associative algebras. Their applications are numerous and well-known in these common settings. We construct P algebra analogues to a series of results from W. R. Scott's Group Theory, which gives an explicit theory of factor systems for the group case. Here P ranges over Leibniz, Zinbiel, diassociative, and dendriform algebras, which we dub "the algebras of Loday," as well as over Lie, associative, and commutative algebras. Fixing a pair of P algebras, we develop a correspondence between factor systems and extensions. This correspondence is strengthened by the fact that equivalence classes of factor systems correspond to those of extensions. Under this correspondence, central extensions give rise to 2-cocycles while split extensions give rise to (nonabelian) 2-coboundaries. [ABSTRACT FROM AUTHOR]