In group theory, a branch of mathematics, the automorphisms and outer automorphisms of the symmetric groups and alternating groups are both standard examples...

alternating group is the group of even permutations of a finite set. The alternating group on a set of n elements is called the alternating group of degree...

outer automorphism of S6—see Automorphisms of the symmetric and alternating groups for details. Note that while A6 and A7 have an exceptional Schur multiplier...

In the mathematical area of group theory, the covering groups of the alternating and symmetric groups are groups that are used to understand the projective...

the subgroup consisting of inner automorphisms. The outer automorphism group is usually denoted Out(G). If Out(G) is trivial and G has a trivial center...

are subrepresentations of the second tensor power. In the language of modules over the group ring, the symmetric and alternating squares are C [ G ] {\displaystyle...

and g is the order of the group of "graph automorphisms" (coming from automorphisms of the Dynkin diagram). The outer automorphism group is often, but...

pi+1. The automorphism groups of p-groups are well studied. Just as every finite p-group has a non-trivial center so that the inner automorphism group is...

simple groups, the only example is in the automorphisms of the symmetric and alternating groups: for n ≥ 3 , n ≠ 6 {\displaystyle n\geq 3,n\neq 6} the alternating...

groups have diagram automorphisms induced by automorphisms of their Dynkin diagrams, and field automorphisms induced by automorphisms of a finite field. Analogously...

18 inner automorphisms. As 2D isometry group D9, the group has mirrors at 20° intervals. The 18 inner automorphisms provide rotation of the mirrors by...

alternating groups (Cameron 1992, p. 139). The only 4-transitive groups are the symmetric groups Sk for k at least 4, the alternating groups Ak for k at...

must be a prime number, and the elementary abelian groups in which the common order is p are a particular kind of p-group. A group for which p = 2 (that...

Cr(Pn(k)) of birational automorphisms; any biregular automorphism is linear, so PGL coincides with the group of biregular automorphisms. Projective transformation...

also the case for the symmetric and alternating square. However, we need a Haar measure on the direct product of compact groups in order to extend the theorem...

isomorphism. The set of all automorphisms of a group G, with functional composition as operation, itself forms a group, the automorphism group of G. It is...

elements of the subgroup. In category theory, quotient groups are examples of quotient objects, which are dual to subobjects. Given a group G {\displaystyle...

and the field automorphism of the extension C/R (namely complex conjugation). Both these automorphisms are automorphisms of the algebraic group, have...

(uncountably) many "wild" automorphisms (assuming the axiom of choice). Field automorphisms are important to the theory of field extensions, in particular...

the orbit-stabilizer theorem to count the automorphisms of a graph. Consider the cubical graph as pictured, and let G denote its automorphism group....

or alternating, or belongs to a broad infinite class called the groups of Lie type, or else it is one of twenty-six exceptions, called sporadic (the Tits...

{\displaystyle A_{n}} – alternating group for n ≥ 5 {\displaystyle n\geq 5} The alternating groups may be considered as groups of Lie type over the field with one...

each of its elements. A general property of finite groups implies that a finite nonempty subset of a symmetric group is a permutation group if and only...

understood. The classification of finite simple groups says that most finite simple groups arise as the group G(k) of k-rational points of a simple algebraic...

a group, called the symmetry group of the object, and the transformations of a given type form a general group. Lie groups appear in symmetry groups in...

considered in mathematics, after cyclic, symmetric and alternating groups, with the projective special linear groups over prime finite fields, PSL(2, p) being...

Alternating group Automorphisms of the symmetric and alternating groups Block (permutation group theory) Cayley's theorem Cycle index Frobenius group...

{\displaystyle \operatorname {GL} (n,F)} and its subgroups are often called linear groups or matrix groups (the automorphism group GL ⁡ ( V ) {\displaystyle \operatorname...

of prime-power order; it is also known as the basis theorem for finite abelian groups. Moreover, automorphism groups of cyclic groups are examples of...

under the symmetric group, and Y {\displaystyle Y} is the quotient by the symmetric group of the non-excluded n {\displaystyle n} -tuples. Under the dimension...