Cayley’s Theorem is one of the fundamental results in abstract algebra, which shows the deep relationship between the structure of groups and semigroups with other mathematical objects, namely permutation groups and transformation semigroups.
Cayley’s Theorem on Groups
Cayley’s Theorem states that every group G is isomorphic to a subgroup of a permutation group, which is a group consisting of bijective functions (permutations) on the set G itself. In essence, every group can be represented as a permutation group of its elements, meaning that each element g in group G can be associated with a function fg that maps element x in G to gx. The function fg is a bijection, and the set of all functions fg forms a subgroup within the symmetry group SymG. Thus, G is isomorphic to this subgroup.
Cayley’s Theorem on Semigroups
For semigroups, a similar result holds with a more general scope. Every semigroup S can be represented as a subsemigroup of a full transformation semigroup, which is the set of all functions from S to S under the binary operation of function composition. In this case, Cayley’s Theorem states that there is always an injective homomorphism mapping from semigroup S to this full transformation semigroup. If S is not a monoid, an identity element will be added in its construction.
Meaning and Applications
This theorem is important because it connects group and semigroup theory with concrete objects (functions and transformations). The existence of Cayley’s Theorem also motivates the emergence of “Cayley-type theorems” for other algebraic structures. Applications of Cayley’s Theorem include various fields such as cryptography, automata theory, and transformation modeling in computation.
Keywords: Theorem, Cayley, Mathematics
References:
Howie, J.M., 1996, Fundamentals of Semigroup Theory, Oxford University Press.
Jampachon, P., Susanti, Y., Denecke, Y., 2012, Four-part Semigroups – Semigroups of Boolean Operations, Discussiones Mathematicae General Algebra and Applications, 32 (2012) 115–116. (doi:10.7151/dmgaa.1188).
Surodjo, B., Susanti, Y., 2023, Teori Semigrup, UGM Press.
Author: Yeni Susanti