Algebra: Abstract and Modern

7

Group Actions on Sets

7.1 Action of a Group on a Set

7.2 Orbits and Stabilizers

7.3 Certain Counting Techniques

7.4 Cauchy and Sylow Theorems

Before the concept of an abstract group took its present shape, the theory of groups dealt only with permutation groups. Abstract groups were introduced much later in order to focus attention on those properties of permutation groups that concern the resultant composition (the binary operation in the permutation groups) only and do not refer to the set on which the permutations act. However, we have seen that any group can be identified (isomorphic) with a group of permutations on some set. Switching back from the abstract point of view to the concrete case of a permutation group is often useful in ...