An Introduction to the Theory of GroupsThis introductory exposition of group theory by an eminent Russian mathematician is particularly suited to undergraduates, developing material of fundamental importance in a clear and rigorous fashion. The treatment is also useful as a review for more advanced students with some background in group theory. Beginning with introductory examples of the group concept, the text advances to considerations of groups of permutations, isomorphism, cyclic subgroups, simple groups of movements, invariant subgroups, and partitioning of groups. An appendix provides elementary concepts from set theory. A wealth of simple examples, primarily geometrical, illustrate the primary concepts. Exercises at the end of each chapter provide additional reinforcement. |
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a₂ addition table alternating permutation angle arbitrary element associative law axes joining axes of symmetry axis b₁ belongs called centroid coincidence congruence groups consider convince ourselves cube defined definition denote diagonals difference module dihedral group displacements double pyramid elements a₁ equal equation equivalence relation equivalent example finite group form a group given element group axioms group G group of order group of rotations group operation group S3 homomorphic mapping icosahedron infinite cyclic group invariant subgroup inverse elements inverse image isomorphic joining the mid-points kernel Klein's four-group line g mapping f movement multiplication natural number null element obtain octahedron odd permutations one-to-one correspondence P₁ partition permutation group plane prove rational numbers real numbers regular polygons rhombus right cosets rotation group second kind set f(A subgroup of order subset symmetric group tetrahedron Theorem transform triangle u₁ union uniquely determined vertex vertices whole numbers x₁ ακ аз