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

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...

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...

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...

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...

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...

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...

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...

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

GL(n, F) and its subgroups are often called linear groups or matrix groups (the automorphism group GL(V) is a linear group but not a matrix group). These...

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...

groups are the symmetric groups Sk for k at least 4, the alternating groups Ak for k at least 6, and the Mathieu groups M24, M23, M12, and M11. (Cameron...

−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...

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

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

{\displaystyle V} is a normal subgroup of the alternating group A 4 {\displaystyle A_{4}} (and also the symmetric group S 4 {\displaystyle S_{4}} ) on four...

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...

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...

Both these automorphisms are automorphisms of the algebraic group, have order 2, and commute, and the unitary group is the fixed points of the product automorphism...

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

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),...

element of K is a symmetric function of the x i , {\displaystyle x_{i},} and is thus fixed by all these automorphisms. It follows that the Galois group Gal...

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

{\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...

The commutator subgroup of the alternating group A4 is the Klein four group. The commutator subgroup of the symmetric group Sn is the alternating group...

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

automorphisms theorem bounds the order of the group of automorphisms, via orientation-preserving conformal mappings, of a compact Riemann surface of genus...

orthogonal groups over perfect fields are the same as symplectic groups in dimension 2n. In fact the symmetric form is alternating in characteristic 2, and as...

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...