Group action: Encyclopedia II - Group action - Types of actions

Group action - Types of actions

The action of G on X is called

• transitive if for any two x, y in X there exists a g in G such that g·x = y;
  • n-transitive if G acts transitively on Xⁿ.
  • sharply n-transitive if G acts regularly on Xⁿ.
• faithful (or effective) if for any two different g, h in G there exists an x in X such that g·x ≠ h·x
• free if for any two different g, h in G and all x in X we have g·x ≠ h·x
• regular (or simply transitive) if it is both transitive and free; this is equivalent to saying that for any two x, y in X there exists precisely one g in G such that g·x = y.

Every free action on a non-empty set is faithful. A group G acts faithfully on X iff the homomorphism G → Sym(X) has a trivial kernel. Thus, for a faithful action, G is isomorphic to a permutation group on X; specifically, G is isomorphic to its image in Sym(X).

The action of any group G on itself by left multiplication is regular, and thus faithful as well. Every group can, therefore, be embedded in the symmetric group on its own elements, Sym(G) — a result known as Cayley's theorem.

If G does not act faithfully on X, one can easily modify the group to obtain a faithful action. If we define N = {g in G : g·x = x for all x in X}, then N is a normal subgroup of G; indeed, it is the kernel of the homomorphism G → Sym(G). The factor group G/N acts faithfully on X by setting (gN)·x = g·x. The original action of G on X is faithful if and only if N = {e}.

Adapted from the Wikipedia article "Types of actions", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki