Cayley's Theorem states that every group G is isomorphic to a subgroup of the symmetric group acting on G, meaning that any abstract group can be represented as a group of permutations. This theorem highlights the fundamental nature of permutation groups in group theory and provides a concrete way to understand abstract groups through permutation representations.

Cayley's Theorem

every group G

isomorphic to a subgroup

The symmetric group, denoted as S_n, is the group of all permutations of a finite set of n elements, capturing the essence of symmetry through its structure. It plays a fundamental role in group theory, serving as a cornerstone for understanding permutation groups and providing insights into the algebraic properties of symmetries in mathematics and physics.

symmetric group

acting on G

abstract group

group of permutations

fundamental nature of permutation groups

Group theory is a branch of abstract algebra that studies the algebraic structures known as groups, which are sets equipped with an operation that satisfies four fundamental properties: closure, associativity, identity, and invertibility. It provides a unifying framework for understanding symmetry in mathematical objects and has applications across various fields including physics, chemistry, and computer science.

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