ABSTRACT
16.1 Symmetry groups or groups of symmetries of objects, that may be represented by finitely many vertices, are always subgroups of the symmetric group on those vertices i.e. they are naturally subgroups of the group of all possible permutations of a certain set. In fact this property holds for all groups (Cayley's Theorem 16.3)! This makes the symmetric groups into a kind of “generic” groups resulting in the philosophy that anything that can go wrong for a group will go wrong for some symmetric group. Nevertheless it is useful to be able to have a unified way of writing and calculating with permutations using standard notation (16.4). Just by using the cycle decomposition for a permutation, some of its properties become clear (Theorem 16.16, Examples 16.19 and 16.21). Viewing a group as a permutation group of some set is also defining a “representation” of that group. The use of Young Tableau's (16.32) is typical for this representation theoretic aspect, they would reappear in a thorough treatment of representations of the symmetric groups in the sense of Chapter 34, we do not go into that here. Another very important issue is the appearance of a simple group, i.e. the alternating group An for n ≥ 5, as a subgroup of index two in the symmetric group which is moreover the only normal subgroup of Sn (in case n ≥ 5). This will have an effect in Galois Theory (see also Chapter 30) concerning the solution of polynomial equations by radicals i.e. the non-solvability of the general quintic by radicals.
