An Introduction to the Theory of Groups

a₁ a₂ addition table alternating permutation angle arbitrary element associative law axes joining axes of symmetry axis b₁ b₂ belongs called centroid coincidence concept congruence groups conjugate elements consider convince ourselves cube defined definition denote diagonals difference module dihedral group displacements double pyramid equal equation equivalence relation example finite group form a group given element group axioms group G group H(a group of order group of rotations group operation homomorphic mapping icosahedron infinite cyclic group invariant subgroup inverse elements inverse image isomorphic joining the mid-points kernel line g mapping ƒ movement natural number null element obtain octahedron odd permutations one-to-one correspondence P₁ P₂ partition permutation group plane prove rational numbers real numbers regular polygons rhombus right cosets rotation group second kind subgroup of order subset symmetric group tetrahedron theorem transform triangle u₁ union uniquely determined v₁ vertex vertices whole numbers x₁