Groups, Algebras and Identities (original) (raw)
2019, Contemporary Mathematics
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This chapter is a concise mathematical introduction into the algebra of groups. It is build up in the way that definitions are followed by propositions and proofs. The concepts and the terminology introduced here will serve as a basis for the following chapters that deal with group theory in the stricter sense and its application to problems in physics. The mathematical prerequisites are at the bachelor level. 1 This is an Open Access article distributed under the terms of the Creative Commons Attribution-Noncommercial License 3.0, which permits unrestricted use, distribution, and reproduction in any noncommercial medium, provided the original work is properly cited.
A Systematic Approach to Group Properties Using its Geometric Structure
European Journal of Pure and Applied Mathematics, 2020
The algebraic properties of a group can be explored through the relationship among its elements. In this paper, we define the graph that establishes a systematic relationship among the group elements. Let G be a finite group, the order product prime graph of a group G, is a graph having the elements of G as its vertices and two vertices are adjacent if and only if the product of their order is a prime power. We give the general presentation for the graph on dihedral groups and cyclic groups and classify finite dihedral groups and cyclic groups in terms of the order product prime graphs as one of connected, complete, regular and planar. We also obtained some invariants of the graph such as its diameter, girth,independent number and the clique number. Furthermore, we used thevertex-cut of the graph in determining the nilpotency status of dihedralgroups. The graph on dihedral groups is proven to be regular and complete only if the degree of the corresponding group is even prime power a...
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