A

group G acts/operates on a

set S iff there exists a

map G x S -> S with
1 s = s and (g h) s = g (h s) for all s of S, g,h of G.

This is equivalent to the existence of a homomorphism T from G into the permutation group of S.

The action is called effective/faithful iff the kernel of T is {1}.

The action is called transitive iff for any fixed s of S then for any t of S there exists g of G with g s = t.

The set O(s) = { h ¦ h of S with there exists g of G with gs = h} is called the orbit of s for s in S.

Examples: G acts on itself per multiplication (effective, transitive)

G acts on itself per conjugation (not transitive for center not equal 1, not effective for center not equal 1)

G acts on the cosets of any subgroup per multiplication (transitive, usually not effective).