An Introduction to the Theory of Groups
This 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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addition table alternating permutation angle arbitrary element associative law axes joining axes of symmetry axis belongs called centroid coincidence congruence groups conjugate elements consider convince ourselves cube deﬁned deﬁnition denote diagonals dihedral group displacements double pyramid elements a1 elements P0 equal equation equivalence relation equivalent example ﬁg ﬁgure ﬁnd ﬁnite group ﬁrst kind ﬁxed form a group given element group axioms group G group of order group of rotations group operation group S3 homomorphic mapping icosahedron invariant subgroup inverse elements inverse image isomorphic joining the mid-points kernel line g mapping f movement multiplication natural number null element obtain octahedron odd permutations one-to-one correspondence one-to-one mapping partition permutation group plane prove rational numbers real numbers regular polygons rhombus right cosets rotation group second kind set G subgroup H subgroups of order subset symmetric group terminology tetrahedron theorem transform triangle union uniquely determined vertex vertices whole numbers