A categorical theorem on universal objects and its application in abelian group theory and computer science

1992 ◽  
pp. 49-74 ◽  
Author(s):  
Manfred Droste ◽  
Rüdiger Göbel
Keyword(s):  
Group Theory ◽  
Abelian Group ◽  
Computer Science

A Classification Of n-Abelian Groups

1969 ◽  
Vol 21 ◽  
pp. 1238-1244 ◽  
Author(s):  
J. L. Alperin
Keyword(s):  
Group Theory ◽  
Abelian Group ◽  
Recent Result ◽  
Abelian Groups ◽  
The Way

The concept of an abelian group is central to group theory. For that reason many generalizations have been considered and exploited. One, in particular, is the idea of an n-abelian group (6). If n is an integer and n > 1, then a group G is n-abelian if, and only if,(xy)n = xnynfor all elements x and y of G. Thus, a group is 2-abelian if, and only if, it is abelian, while non-abelian n-abelian groups do exist for every n > 2.Many results pertaining to the way in which groups can be constructed from abelian groups can be generalized to theorems on n-abelian groups (1; 2). Moreover, the case of n = p, a prime, is useful in the study of finite p-groups (3; 4; 5). Moreover, a recent result of Weichsel (9) gives a description of all p-abelian finite p-groups.

scholarly journals p-Grup pada Grup Dihedral

2019 ◽  
Author(s):  
Muhammad Irfan Hidayat
Keyword(s):  
Group Theory ◽  
Abelian Group ◽  
Prime Number ◽  
Dihedral Group ◽  
Binary Operations ◽  
Interesting Part ◽  
Over Time

Group theory is an interesting part of algebra. The group theory is often researched and developed over time. The group is defined as a set with binary operations and fulfills several other conditions. One interesting and often discussed group is the Dihedral Group. The Dihedral group denoted by D_2n is the set of regular n-aspect symmetries, ∀nϵN, n≥3 with the composition operation "◦" which satisfies the axioms of the group and does not belong to the abelian group (commutative) while the form of the group is D_2n = {e, a, a ^ 2, ..., a ^ (n-1), b, ab, a ^ 2 b, .., a ^ (n-1) b} with n≥3. From a D_2n group subgroups can be formed which can also be viewed as another group. This research will examine several subgroups of group D_2n which are p-groups. p-Group is a group with the order p where p is a prime number. Previously all forms of subgroups had been obtained from the Dihedral group (D_2n). Based on that, this research will look for the D_2n subgroup that forms the p-group by identifying orders from the dihedral group.

scholarly journals Groups – Additive Notation

2015 ◽  
Vol 23 (2) ◽  
pp. 127-160 ◽  
Author(s):  
Roland Coghetto
Keyword(s):  
Topological Group ◽  
Group Theory ◽  
Abelian Group ◽  
Normal Subgroup ◽  
Dense Subset ◽  
Abelian Topological Group ◽  
Covering Group ◽  
Mizar Mathematical Library ◽  
Order Of An Element ◽  
Generic Theory

Abstract We translate the articles covering group theory already available in the Mizar Mathematical Library from multiplicative into additive notation. We adapt the works of Wojciech A. Trybulec [41, 42, 43] and Artur Korniłowicz [25]. In particular, these authors have defined the notions of group, abelian group, power of an element of a group, order of a group and order of an element, subgroup, coset of a subgroup, index of a subgroup, conjugation, normal subgroup, topological group, dense subset and basis of a topological group. Lagrange’s theorem and some other theorems concerning these notions [9, 24, 22] are presented. Note that “The term ℤ-module is simply another name for an additive abelian group” [27]. We take an approach different than that used by Futa et al. [21] to use in a future article the results obtained by Artur Korniłowicz [25]. Indeed, Hölzl et al. showed that it was possible to build “a generic theory of limits based on filters” in Isabelle/HOL [23, 10]. Our goal is to define the convergence of a sequence and the convergence of a series in an abelian topological group [11] using the notion of filters.

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