Algebra: Abstract and Modern

6

Permutation Groups

6.1 Cayley’s Theorem

6.2 The Symmetric Group S_n

6.3 Cycles

6.4 Alternating Group A_n and Dihedral Group D_n

For any nonempty set X, the set M(X) of all mappings of X into itself forms a monoid under the composition of mappings in which the invertible elements (elements possessing inverses) are precisely the bijections of X onto itself. In fact, we have observed that a mapping f : X → X has left (right) inverse in M(X) if and only if f is an injection (respectively, surjection) and, as such, the set of all bijections of X onto itself forms a group under the composition of mappings. Before the advent of the abstract form of a group, mathematicians were only interested in the group structure of certain sets of bijections, which ...