Representation Theory/Set representations

set representation A set representation is a representation of a group in the category of sets. Alternatively, set representations are also called group actions, and we say that acts on a set . Whenever , we will denote the corresponding element of (which are just the permutations of ) by as well, so that becomes a bijective function on . In particular, for , we can make sense of expressions such as (which shall be a shorthand for ). orbit Let be a group acting on a set , and let . The orbit of the element is defined to be the set . group action partitions set into orbits Let be a group acting on a set . Then is partitioned into orbits of . We prove the claim by proving that being in the same orbit is an equivalence relation. , since as the representation is a group homomorphism, so that represents the identity since as the representation is a group morphism, so that since and imply since as the representation is a group homomorphism transitive action Suppose that is a group acting on . This action is called transitive if and only if all elements are in the same orbit, ie. there is only one orbit, and it is all of . Equivalently, we could have required that for all , there exists such that .