Groups: Introductory Concepts

Mahima Ranjan Adhikari and Avishek Adhikari
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Mahima Ranjan Adhikari: Institute for Mathematics, Bioinformatics, Information Technology and Computer Science (IMBIC)
Avishek Adhikari: University of Calcutta, Department of Pure Mathematics

Chapter Chapter 2 in Basic Modern Algebra with Applications, 2014, pp 55-136 from Springer

Abstract: Abstract Chapter 2 gives an introduction to the group theory. This concept is used in subsequent chapters. Groups serve as one of the fundamental building blocks for the subject called today modern algebra. The theory of groups began with the work of J.L. Lagrange (1736–1813) and E. Galois (1811–1832). At that time, mathematicians worked with groups of transformations. These were sets of mappings which possessed certain properties under composition. Mathematicians, such as Felix Klein (1849–1925) adopted the idea of groups to unify different areas of geometry. In 1870, L. Kronecker (1823–1891) gave a set of postulates for a group. Earlier definitions of groups were generalized to the present concept of an abstract group in the first decade of the twentieth century, which was defined by a set of axioms. In this chapter we make an introductory study of groups with geometrical applications along with a discussion on free abelian groups and structure theorem for finitely generated abelian groups. Moreover, semigroups, homology groups, cohomology groups, topological groups, Lie groups, Hopf’s groups and fundamental groups are studied here.

Keywords: Finitely-generated Abelian Group; Inverse Semigroup; Regular Semigroup; Internal Direct Product; Hopf Group (search for similar items in EconPapers)
Date: 2014
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DOI: 10.1007/978-81-322-1599-8_2

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