A generalization of groups

European Journal of Pure and Applied Mathematics

A nonempty set G is a g-group [with respect to a binary operation ∗] if it satisfies the following properties: (g1) a ∗ (b ∗ c) = (a ∗ b) ∗ c for all a, b, c ∈ G; (g2) for each a ∈ G, there exists an element e ∈ G such that a ∗ e = a = e ∗ a (e is called an identity element of a); and, (g3) for each a ∈ G, there exists an element b ∈ G such that a ∗ b = e = b ∗ a for some identity element eof a. In this study, we gave some important properties of g-subgroups, homomorphism of g-groups, andthe zero element. We also presented a couple of ways to construct g-groups and g-subgroups.

2006

Journal of Pure and Applied Algebra, 1983

Chinese Science Bulletin, 1997

Proceedings of the American Mathematical Society, 2017

We study two complexity notions of groups-the syntactic complexity of a computable Scott sentence and the m-degree of the index set of a group. Finding the exact complexity of one of them usually involves finding the complexity of the other, but this is not always the case. Knight et al. determined the complexity of index sets of various structures. In this paper, we focus on finding the complexity of computable Scott sentences and index sets of various groups. We give computable Scott sentences for various different groups, including nilpotent groups, polycyclic groups, certain solvable groups, and certain subgroups of Q. In some of these cases, we also show that the sentences we give are optimal. In the last section, we also show that d-Σ 2 Δ 3 in the complexity hierarchy of pseudo-Scott sentences, contrasting the result saying d-Σ 2 = Δ 3 in the complexity hierarchy of Scott sentences, which is related to the boldface Borel hierarchy.

Journal of University of Anbar for Pure Science

The purpose of this paper is to study the concept of dependence , independence and the basis of some algebraic structure and give the definition of a finite group with basic property and study some of its basic properties .

Groups St Andrews 2001 in Oxford, 2000

This paper is concerned with finite groups G(•) and G(*) of order n that are not isomorphic, and where the size of {(u, v) ∈ G × G; u • v = u * v} is the least possible (with respect to the given n). It surveys the case of 2-groups, discusses the possible generalization of the known results to p-groups, p an odd prime, and establishes the least possible distance in the case when G(*) is an elementary abelian 3-group. Let G(•) and G(*) be finite groups of order n. Consider the set {(u, v) ∈ G × G; u • v = u * v} and denote its size by d(•, *). The number d(•, *) is called the (Hamming) distance of • and *. If d(•, *) < n 2 /4 and n is a power of two, then G(•) ∼ = G(*), by [4]. Section 1 lists further results about distances of 2groups, while Section 2 discusses the associated proof machinery, and its possible generalization to p-groups, p an odd prime. In Section 2 there are also presented non-isomorphic p-groups G(•) and G(*), |G| = n > p, for which d(•, *) = n 2 (p 2 − 1)/(4p 2). This result is the best possible, when G(•) is an elementary abelian 3-group-in Section 3 we shall show that in such a case d(•, *) < 2n 2 /9 implies G(•) ∼ = G(*). If H ≤ G(•), then the set of all left (or right) cosets of H in G(•) is denoted by L • (H) and R • (H), respectively. If A ⊆ G and B ⊆ G, then the size of {(u, v) ∈ A × B; u • v = u * v} will be denoted by d(A, B).

Archiv der Mathematik, 1985

Obviously P2 is commutativity, while P3, P4, etc. are successively weaker properties. A group will be said to have the property P if it satisfies P, for some n > 1. The property P is a finiteness condition in the sense of ; indeed every finite group of order m trivially has Pro" The study of the property P was initiated in [1], while the corresponding property for semigroups has been considered in . The object of this note is to give a complete classification of groups with the property P. Our main result is It was shown in [1] that every P-group is FC-nilpotent. This result can now be sharpened since it follows from the Theorem (or Lemma (2.1)) that the FC-centre of a P-group has finite index. The FC-nilpotent class is therefore at most 2; in fact the class will be < 2 precisely when the group has finite derived subgroup (see (4.1)). 2. Preliminary results. The proof of the Theorem depends on the following key lemma.

TURKISH JOURNAL OF MATHEMATICS

A group is said to satisfy a word w in the symbols {x, x −1 , y, y −1 } provided that if the 'x' and 'y' are replaced by arbitrary elements of the group then the equation w = 1 is satisfied. This paper studies certain equations in words, as above, which together with other conditions imply that groups which satisfy these equations and conditions must be abelian.