FRUCHT S THEOREM
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In the mathematical discipline of group theory, Cayley's theorem, named in honour of Arthur Cayley, states that every group G is isomorphic to a subgroup of a symmetric group. More specifically, G is isomorphic to a subgroup of the symmetric group Sym ( G ) {\displaystyle \operatorname {Sym} (G)} whose elements are the permutations of the underlying set of G. Explicitly, for each g ∈ G {\displaystyle g\in G} , the left-multiplication-by-g map ℓ g : G → G {\displaystyle \ell _{g}\colon G\to G} sending each element x to gx is a permutation of G, and the map G → Sym ( G ) {\displaystyle G\to \operatorname {Sym} (G)} sending each element g to ℓ g {\displaystyle \ell _{g}} is an injective homomorphism, so it defines an isomorphism from G onto a subgroup of Sym ( G ) {\displaystyle \operatorname {Sym} (G)} . The homomorphism G → Sym ( G ) {\displaystyle G\to \operatorname {Sym} (G)} can also be understood as arising from the left translation action of G on the underlying set G. When G is finite, Sym ( G ) {\displaystyle \operatorname {Sym} (G)} is finite too. The proof of Cayley's theorem in this case shows that if G is a finite group of order n, then G is isomorphic to a subgroup of the standard symmetric group S n {\displaystyle S_{n}} . But G might also be isomorphic to a subgroup of a smaller symmetric group, S m {\displaystyle S_{m}} for some m < n {\displaystyle m
In connection with: Cayley's theorem
Title combos: theorem Cayley
Description combos: in displaystyle so groups are to defines the sending
This is a list of notable theorems. Lists of theorems and similar statements include: List of algebras List of algorithms List of axioms List of conjectures List of data structures List of derivatives and integrals in alternative calculi List of equations List of fundamental theorems List of hypotheses List of inequalities Lists of integrals List of laws List of lemmas List of limits List of logarithmic identities List of mathematical functions List of mathematical identities List of mathematical proofs List of misnamed theorems List of scientific laws List of theories Most of the results below come from pure mathematics, but some are from theoretical physics, economics, and other applied fields.
In connection with: List of theorems
Title combos: theorems of of theorems List
Description combos: from inequalities of of List laws axioms of of
Frucht is surname of: Adolf-Henning Frucht (1913, Torgau - 1993), German doctor and physiologist Michael Frucht (1969-), an American neurologist [2] Robert (Roberto) Wertheimer Frucht (1906 - 1997), a German-Chilean mathematician Frucht graph Frucht's theorem
In connection with: Frucht
Description combos: Frucht physiologist Frucht Michael American Henning German 1906 theorem
In graph theory, a branch of mathematics, an undirected graph is called an asymmetric graph if it has no nontrivial symmetries. Formally, an automorphism of a graph is a permutation p of its vertices with the property that any two vertices u and v are adjacent if and only if p(u) and p(v) are adjacent. The identity mapping of a graph is always an automorphism, and is called the trivial automorphism of the graph. An asymmetric graph is a graph for which there are no other automorphisms. Note that the term "asymmetric graph" is not a negation of the term "symmetric graph," as the latter refers to a stronger condition than possessing nontrivial symmetries.
In connection with: Asymmetric graph
Title combos: graph Asymmetric
Description combos: permutation nontrivial any called if branch theory graph stronger
In the mathematical field of graph theory, the Frucht graph is a cubic graph with 12 vertices, 18 edges, and no nontrivial symmetries. It was first described by Robert Frucht in 1949. The Frucht graph is a pancyclic, Halin graph with chromatic number 3, chromatic index 3, radius 3, and diameter 4. Like every Halin graph, the Frucht graph is polyhedral (planar and 3-vertex-connected) and Hamiltonian, with girth 3. Its independence number is 5. The Frucht graph can be constructed from the LCF notation: [−5,−2,−4,2,5,−2,2,5,−2,−5,4,2].
In connection with: Frucht graph
Title combos: graph Frucht
Description combos: Like 18 The and with Robert of in 1949
Frucht's theorem is a result in algebraic graph theory, conjectured by Dénes Kőnig in 1936 and proved by Robert Frucht in 1939. It states that every finite group is the group of symmetries of a finite undirected graph. More strongly, for any finite group G there exist infinitely many non-isomorphic simple connected graphs such that the automorphism group of each of them is isomorphic to G.
In connection with: Frucht's theorem
Title combos: theorem Frucht
Description combos: of by connected isomorphic the group in isomorphic each
Robert Wertheimer Frucht (later known as Roberto Frucht) (9 August 1906 – 26 June 1997) was a German-Chilean mathematician; his research specialty was graph theory and the symmetries of graphs.
In connection with: Robert Frucht
Title combos: Frucht Robert
Description combos: graphs his Roberto 26 and of later June August
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