Cayley's theorem

Every group is isomorphic to a subgroup of a symmetric group. The embedding sends each group element to the permutation of the group given by left multiplication.

Group action: A group action assigns each group element a transformation of a set while respecting identity and multiplication. Left multiplication is the action that produces the permutations in Cayley's construction.
Group: A set with an associative binary operation, an identity element, and an inverse for every element. The theorem applies to every structure satisfying these group axioms.
Arthur Cayley: A British mathematician whose work helped establish the modern abstract theory of groups and matrices. Cayley gave the theorem its name through his work on abstract groups.
Cayley graph: A graph encoding a group's generators and multiplication through edges between group elements. The same regular action underlying Cayley's theorem organizes group elements into a graph.
Left regular representation: The representation of a group by linear operators obtained from its action on itself by left multiplication. It is the linear form of the same faithful action used in the theorem.
Symmetric group: The group of all permutations of a set, with composition as its operation. The theorem embeds a group into the symmetric group on its own elements.
Abstract group: A group studied through its axioms and operation rather than as a particular set of transformations. The theorem shows every abstract group can be realized concretely as permutations.
Coset action: The action of a group on the left cosets of a subgroup by left multiplication. It generalizes the theorem's action on the group to actions on coset spaces.
Faithful group action: A group action in which distinct group elements induce distinct permutations of the acted-on set. Faithfulness is exactly the injectivity needed for the map into the symmetric group.
Group homomorphism: A function between groups that preserves the group operation. The left-multiplication assignment is a homomorphism because composing its permutations matches multiplication.

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