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...

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...

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...

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...

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

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...

−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...

mathematics, the category Grp (or Gp) has the class of all groups for objects and group homomorphisms for morphisms. As such, it is a concrete category. The study...

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

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...

called symmetric if it stays the same whenever the indeterminates are permuted in any way. More formally, there is an action by ring automorphisms of the symmetric...

{\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...

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...

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

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

Hilbert showed that all symmetric and alternating groups are represented as Galois groups of polynomials with rational coefficients. The polynomial xn + ax...

important in Bass–Serre theory, the study of groups acting by automorphisms on trees. Specifically, any group acting with finite vertex stabilizers on...

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

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...