Search results
Results from the WOW.Com Content Network
The center of the symmetric group, S n, is trivial for n ≥ 3. The center of the alternating group, A n, is trivial for n ≥ 4. The center of the general linear group over a field F, GL n (F), is the collection of scalar matrices, { sI n ∣ s ∈ F \ {0} }. The center of the orthogonal group, O n (F) is {I n, −I n}.
For n = 2, the automorphism group is trivial, but S 2 is not trivial: it is isomorphic to C 2, which is abelian, and hence the center is the whole group. For n = 6 , it has an outer automorphism of order 2: Out(S 6 ) = C 2 , and the automorphism group is a semidirect product Aut(S 6 ) = S 6 ⋊ C 2 .
Every symmetric group has a one-dimensional representation called the trivial representation, where every element acts as the one by one identity matrix. For n ≥ 2 , there is another irreducible representation of degree 1, called the sign representation or alternating character , which takes a permutation to the one by one matrix with entry ...
The alternating group, symmetric group, and their double covers are related in this way, and have orthogonal representations and covering spin/pin representations in the corresponding diagram of orthogonal and spin/pin groups. Explicitly, S n acts on the n-dimensional space R n by permuting coordinates (in matrices, as permutation matrices).
In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the ambient space which takes the object to itself, and which preserves all the relevant structure of the object.
A subgroup H of a group G is called a self-normalizing subgroup of G if N G (H) = H. The center of G is exactly C G (G) and G is an abelian group if and only if C G (G) = Z(G) = G. For singleton sets, C G (a) = N G (a). By symmetry, if S and T are two subsets of G, T ⊆ C G (S) if and only if S ⊆ C G (T).
As an example, all the symmetric groups, S n, are complete except when n ∈ {2, 6}. For the case n = 2, the group has a non-trivial center, while for the case n = 6, there is an outer automorphism. The automorphism group of a simple group is an almost simple group; for a non-abelian simple group G, the automorphism group of G is complete.
In mathematics, the outer automorphism group of a group, G, is the quotient, Aut(G) / Inn(G), where Aut(G) is the automorphism group of G and Inn(G) is 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, then G is said to be complete.