Hhatás

Hhatás, often written as H-hatás in Hungarian mathematical literature, denotes the action of a group H on a set X. An action is a map •: H × X → X, usually written h • x, that satisfies two axioms: the identity element e of H acts as the identity on X, e • x = x for all x ∈ X; and compatibility, h1 • (h2 • x) = (h1 h2) • x for all h1, h2 ∈ H and x ∈ X. These conditions mean that H acts by transformations of X.

For x ∈ X, the orbit Hx = {h • x | h ∈ H} is the set of elements reachable from x by the action. The stabilizer Hx = {h ∈ H | h • x = x} is the subgroup fixing x. Orbits partition X. If H is finite, the orbit-stabilizer theorem states that |Orbit(x)| = |H| / |Stab_H(x)|, so the size of an orbit is the index of the stabilizer.

Examples include the left regular action of H on itself, where h • g = h g; the action of the symmetric group Sn on the set {1, ..., n} by permutation; and the natural linear action of GL(V) on a vector space V, where g • v = gv. The kernel of an action is ker = {h ∈ H | h • x = x for all x ∈ X}; the action is faithful when ker is trivial.

The set X modulo the action, denoted X/H, consists of the orbits. Invariants are elements fixed by all elements of H. Group actions are fundamental in areas such as geometry, combinatorics, and representation theory, where they describe symmetries, quotient spaces, and symmetry-related classifications.