Group Action - Types of Actions

Types of Actions

The action of G on X is called

• Transitive if X is non-empty and if for any x, y in X there exists a g in G such that g.x = y.
• Faithful (or effective) if for any two distinct g, h in G there exists an x in X such that g.x ≠ h.x; or equivalently, if for any g≠ e in G there exists an x in X such that g.x ≠ x. Intuitively, in a faithful group action, different elements of G induce different permutations of X.
• Free (or semiregular) if, given g, h in G, the existence of an x in X with g.x = h.x implies g = h. Equivalently: if g is a group element and there exists an x in X with g.x = x (that is, if g has at least one fixed point), then g is the identity.
• Regular (or simply transitive or sharply 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. In this case, X is known as a principal homogeneous space for G or as a G-torsor.
• n-transitive if X has at least n elements and for any pairwise distinct x₁, ..., x_n and pairwise distinct y₁, ..., y_n there is a g in G such that g·x_k = y_k for 1 ≤ k ≤ n. A 2-transitive action is also called doubly transitive, a 3-transitive action is also called triply transitive, and so on. Such actions define 2-transitive groups, 3-transitive groups, and multiply transitive groups.
  • Sharply n-transitive if there is exactly one such g. See also sharply triply transitive groups.
• Primitive if it is transitive and preserves no non-trivial partition of X. See primitive permutation group for details.
• Locally free if G is a topological group, and there is a neighbourhood U of e in G such that the restriction of the action to U is free; that is, if g.x = x for some x and some g in U then g = e.
• Irreducible if X is a non-zero module over a ring R, the action of G is R-linear, and there is no nonzero proper invariant submodule.

Every free action on a non-empty set is faithful. A group G acts faithfully on X if and only if the corresponding 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(X). 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}.