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...

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...

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...

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...

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...

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...

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

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...

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

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

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...

−1}. The center of the symmetric group, Sn, is trivial for n ≥ 3. The center of the alternating group, An, is trivial for n ≥ 4. The center of the general...

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

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...

subset of a symmetric group that is closed under composition of permutations, contains the identity permutation, and contains the inverse permutation of each...

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

{\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...

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

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...

linear automorphisms is instead the order 2 extension PGL(2, 7), and the group of collineations of the projective line is the complete symmetric group of the...

branch of mathematics, a skew-symmetric graph is a directed graph that is isomorphic to its own transpose graph, the graph formed by reversing all of its...

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

cyclic groups. Every cyclic group of prime order is a simple group, which cannot be broken down into smaller groups. In the classification of finite simple...

the alternating groups of degree at least 5, the infinite family of commutator groups 2F4(22n+1)′ of groups of Lie type (containing the Tits group),...

Buchsbaum, D., Groups, Rings, Modules (Mineola, NY: Dover Publications, 1974), pp. 28–29. Stanojkovski, M., Intense Automorphisms of Finite Groups (Providence...