On the theory of groups, as depending on the symbolic equation theta^n = 1¶
Cayley, A. (1854). On the theory of groups, as depending on the symbolic equation theta^n = 1. Philosophical Magazine, 7(42), 40-47.
Cited by¶
3 citations across 3 artifacts.
Each citation links to the sentence it supports in the citing article.
Primes¶
- Associativity
- Associativity is the regrouping-without-effect principle that: (1) for a binary operation \(\circ\) on a set, the result of combining three or more elements does not depend on how they are grouped (parenthesized) — formally, \((a \circ b) \circ c = a \circ (b \circ c)\) for all elements; the consequence is that a finite sequence of operations has an unambiguous result regardless of evaluation grouping, so expressions can be written without explicit parentheses or grouping markers once the operation is known to associate; this principle emerged explicitly in 19-century algebra when Cayley (1854) gave the first abstract definition of a group as a set with an associative binary operation, identity, and inverses, codifying closure and associativity as foundational axioms.
This sourceThe first abstract definition of a group as a set closed under an associative binary operation with identity and inverses
- Associativity is the regrouping-without-effect principle that: (1) for a binary operation \(\circ\) on a set, the result of combining three or more elements does not depend on how they are grouped (parenthesized) — formally, \((a \circ b) \circ c = a \circ (b \circ c)\) for all elements; the consequence is that a finite sequence of operations has an unambiguous result regardless of evaluation grouping, so expressions can be written without explicit parentheses or grouping markers once the operation is known to associate; this principle emerged explicitly in 19-century algebra when Cayley (1854) gave the first abstract definition of a group as a set with an associative binary operation, identity, and inverses, codifying closure and associativity as foundational axioms.
- Isomorphism
- … set with an associative binary operation, an identity, and inverses) and establishes that every group is isomorphic to a subgroup of a symmetric group via the regular representation — a result now known as Cayley's theorem and an early structural-classification result that uses isomorphism as its organising notion.
This source(First abstract definition of a group as a set with an associative binary operation, an identity, and inverses; statement and proof of Cayley's theorem on the regular representation, establishing that every group is isomorphic to a subgroup of a symmetric group on its underlying set.)
- … set with an associative binary operation, an identity, and inverses) and establishes that every group is isomorphic to a subgroup of a symmetric group via the regular representation — a result now known as Cayley's theorem and an early structural-classification result that uses isomorphism as its organising notion.
Mechanisms¶
- Permutation Group Model
- It is the right model whenever the objects of interest are rearrangements — shuffles, wiring permutations, ringing changes — and Cayley's theorem
This sourceShows that every finite group can be represented as a group of permutations.
- It is the right model whenever the objects of interest are rearrangements — shuffles, wiring permutations, ringing changes — and Cayley's theorem
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