Group Action

• Let be a set and a group. An action of on is a map such that and

• in the above is called the -set.

• (Fraleigh 16.3) Let be a -set.

  For each , the function . defined as

  is a permutation of .

  Also, the map defined by is a Group Homomorphism with the property that

  • Intuition: We establish that is a one-to-one map of onto itself, which proves that it is a permutation, which follows from the two properties of the group action.

    The homomorphism follows by showing that maps to the same element as satisfying the homomorphism property. This immediately follows from the composition of permutations.

  • is transitive on -set if and only if is transitive.

• The subset of leaving every element of fixed is a Normal Group .

• acts faithfully on if only the identity element leaves every fixed.

• Two -sets are isomorphic if there exists a bijective mapping such that

• Let be a -set and . The subgroup is the isotropy subgroup or stabilizer of defined as

  See (Fraleigh 16.12) for why it is a Subgroup.

• The orbit of under is defined as the partition of the equivalence relation defined where:

  We denote the orbit as

  • In other words, the orbit is the set of all elements in reached by repeatedly applying the group action to .
  • See (Fraleigh 16.14) for why it is an equivalence Relation.
  • (Fraleigh e16.6) Every -set is the union of its orbits. Also, the union of -sets is a -set.

• (Fraleigh 16.16) Orbit-Stabilizer Theorem Let be a -set and . Then

  Also if is finite,

  • Intuition: We can establish a one-to-one map from to the collection of left cosets of which gives the first relation. The second relation follows immediately from the definition of the group index.
  • Intuition A second way to view this is with the lens of the fundamental homomorphism theorem. Define a homomorphism in the obvious way using the group operation on . The elements of form a -set. It can then be shown that and . The Orbit-Stabilizer theorem immediately follows.

• (Fraleigh e16.16) Every -set is isomorphic to a disjoint union of left coset -sets.

• (Fraleigh 17.1) Burnside’s Lemma. Let be a finite group and a finite -set. Then the number of orbits is calculated as

  Where denotes the elements of fixed by . In other words the number of orbits, equals the average number of fixed points.

  • Proof: The lemma follows by counting two ways in the following equation

    Both sides count how many total fixed points are possible. That is, the number of pairs where .

    The RHS counts this on , counting the fixed points for each element

    The LHS counts this on can be better expressed as follows. The number is precisely by . The Orbit-Stabilizer Theorem then gives the following.

    Let be the set of unique orbits. Let . We can group each that are part of the same orbit (since orbits define an equivalence relation).

Links §

• Cosets, Group Indices
• A First Course in Abstract Algebra 7th Edition by Fraleigh