Cayley\'s theorem - Citizendia

In group theory, Cayley's theorem, named in honor of Arthur Cayley, states that every group G is isomorphic to a subgroup of the symmetric group on G. Group theory is a mathematical discipline the part of Abstract algebra that studies the Algebraic structures known as groups. Arthur Cayley ( August 16 1821 - January 26 1895) was a British Mathematician. In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element In Abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in In Group theory, given a group G under a Binary operation * we say that some Subset H of G is a subgroup of In Mathematics, the symmetric group on a set X, denoted by S X or Sym( X) is the group whose underlying This can be understood as an example of the group action of G on the elements of G. In Algebra and Geometry, a group action is a way of describing symmetries of objects using groups.

A permutation of a set G is any bijective function taking G onto G; and the set of all such functions forms a group under function composition, called the symmetric group on G, and written as Sym(G). In several fields of Mathematics the term permutation is used with different but closely related meanings In Mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function In Mathematics, a composite function represents the application of one function to the results of another

Cayley's theorem puts all groups on the same footing, by considering any group (including infinite groups such as (R,+)) as a permutation group of some underlying set. In Mathematics, a permutation group is a group G whose elements are Permutations of a given set M, and whose group operation Thus, theorems which are true for permutation groups are true for groups in general.

1 History
2 Proof of the theorem
- 2.1 Alternate setting of proof
3 Remarks on the regular group representation
4 Examples of the regular group representation
5 References
6 See also

History

Attributed in Burnside^[1] to Jordan^[2], Eric Nummela^[3] nonetheless argues that the standard name for this theorem -- "Cayley's Theorem" -- is in fact appropriate. Cayley, in his original 1854 paper^[4] in which introduced the concept of a group, showed that the correspondence in the theorem is one-to-one, but he failed to explicitly show it was a homomorphism (and thus an isomorphism). However, Nummela notes that Cayley made this result known to the mathematical community at the time, thus predating Jordan by 16 years or so.

Proof of the theorem

From elementary group theory, we can see that for any element g in G, we must have g*G = G; and by cancellation rules, that g*x = g*y if and only if x = y. In Mathematics, a group G,*> is defined as a set G and a Binary operation on G, called product and denoted So multiplication by g acts as a bijective function f_g : G → G, by defining f_g(x) = g*x. In Mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property Thus, f_g is a permutation of G, and so is a member of Sym(G).

The subset K of Sym(G) defined as K = {f_g : g in G and f_g(x) = g*x for all x in G} is a subgroup of Sym(G) which is isomorphic to G. The fastest way to establish this is to consider the function T : G → Sym(G) with T(g) = f_g for every g in G. T is a group homomorphism because (using "•" for composition in Sym(G)):(f_g • f_h)(x) = f_g(f_h(x)) = f_g(h*x) = g*(h*x) = (g*h)*x = f_(g*h)(x), for all x in G, and hence: T(g) • T(h) = f_g • f_h = f_(g*h) = T(g*h). In Mathematics, given two groups ( G, * and ( H, · a group homomorphism from ( G, * to ( H, · is a function The homomorphism T is also injective since T(g) = id_G (the identity element of Sym(G)) implies that g*x = x for all x in G, and taking x to be the identity element e of G yields g = g*e = e.

Thus G is isomorphic to the image of T, which is the subgroup K considered earlier.

T is sometimes called the regular representation of G.

Alternate setting of proof

An alternate setting uses the language of group actions. In Algebra and Geometry, a group action is a way of describing symmetries of objects using groups. We consider the group $G$ as a G-set, which can be shown to have permutation representation, say $φ$ .

Firstly, suppose $G = G / H$ with $H = {e}$ . Then the group action is $g . e$ by classification of G-orbits (also known as the orbit-stabilizer theorem). In Algebra and Geometry, a group action is a way of describing symmetries of objects using groups.

Now, the representation is faithful if $φ$ is injective, that is, if the kernel of $φ$ is trivial. Suppose $g$ ∈ ker $φ$ Then, $g = g . e = φ(g). e$ by the equivalence of the permutation representation and the group action. But since $g$ ∈ ker $φ$ , $φ(g) = e$ and thus ker $φ$ is trivial. Then im $φ < G$ and thus the result follows by use of the first isomorphism theorem. In Mathematics, the isomorphism theorems are three Theorems applied widely in the realm of Universal algebra, stating the existence of certain Natural

Remarks on the regular group representation

The identity group element corresponds to the identity permutation. All other group elements correspond to a permutation that does not leave any element unchanged. Since this also applies for powers of a group element, lower than the order of that element, each element corresponds to a permutation which consists of cycles which are of the same length: this length is the order of that element. The elements in each cycle form a left coset of the subgroup generated by the element. In Mathematics, if G is a group, H is a Subgroup of G, and g is an element of G, then gH

Examples of the regular group representation

Z₂ = {0,1} with addition modulo 2; group element 0 corresponds to the identity permutation e, group element 1 to permutation (12).

Z₃ = {0,1,2} with addition modulo 3; group element 0 corresponds to the identity permutation e, group element 1 to permutation (123), and group element 2 to permutation (132). E. g. 1 + 1 = 2 corresponds to (123)(123)=(132).

Z₄ = {0,1,2,3} with addition modulo 4; the elements correspond to e, (1234), (13)(24), (1432).

The elements of Klein four-group {e, a, b, c} correspond to e, (12)(34), (13)(24), and (14)(23). In Mathematics, the Klein four-group (or just Klein group or Vierergruppe, often symbolized by the letter V) is the group Z2

S₃ (dihedral group of order 6) is the group of all permutations of 3 objects, but also a permutation group of the 6 group elements:

*	e	a	b	c	d	f	permutation
e	e	a	b	c	d	f	e
a	a	e	d	f	b	c	(12)(35)(46)
b	b	f	e	d	c	a	(13)(26)(45)
c	c	d	f	e	a	b	(14)(25)(36)
d	d	c	a	b	f	e	(156)(243)
f	f	b	c	a	e	d	(165)(234)

References

^ Burnside, William (1911), Theory of Groups of Finite Order (2 ed. The smallest Non-abelian group has 6 elements It is a Dihedral group with notation D 3 or D 6 William Burnside ( July 2 1852 - August 21 1927) was an English Mathematician. ), Cambridge
^ Jordan, Camille (1870), Traite des substitutions et des equations algebriques, Paris: Gauther-Villars
^ Nummela, Eric (1980), “Cayley's Theorem for Topological Groups”, American Mathematical Monthly 87 (3): 202-203
^ Cayley, Arthur (1854), “On the theory of groups as depending on the symbolic equation θⁿ=1”, Phil. Marie Ennemond Camille Jordan ( January 5 1838 &ndash January 22 1922) was a French Mathematician, known both for his foundational Arthur Cayley ( August 16 1821 - January 26 1895) was a British Mathematician. Mag. 7 (4): 40-47

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