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A group G is said to be linear if there exists a field K, an integer d and an injective homomorphism from G to the general linear group GL d (K) (a faithful linear representation of dimension d over K): if needed one can mention the field and dimension by saying that G is linear of degree d over K.
In all these cases except for D 4, there is a single non-trivial automorphism (Out = C 2, the cyclic group of order 2), while for D 4, the automorphism group is the symmetric group on three letters (S 3, order 6) – this phenomenon is known as "triality". It happens that all these diagram automorphisms can be realized as Euclidean symmetries ...
A 1 (2) is isomorphic to the symmetric group on 3 points of order 6. A 1 (3) is isomorphic to the alternating group A 4 (solvable). A 1 (4) and A 1 (5) are both isomorphic to the alternating group A 5. A 1 (7) and A 2 (2) are isomorphic. A 1 (8) is isomorphic to the derived group 2 G 2 (3)′. A 1 (9) is isomorphic to A 6 and to the derived ...
The infinite alternating group , i.e. the group of even finitely supported permutations of the integers, is simple. This group can be written as the increasing union of the finite simple groups A n {\displaystyle A_{n}} with respect to standard embeddings A n → A n + 1 {\displaystyle A_{n}\rightarrow A_{n+1}} .
In mathematics, a Young tableau (/ t æ ˈ b l oʊ, ˈ t æ b l oʊ /; plural: tableaux) is a combinatorial object useful in representation theory and Schubert calculus.It provides a convenient way to describe the group representations of the symmetric and general linear groups and to study their properties.
In mathematics, the general linear group of degree n is the set of n×n invertible matrices, together with the operation of ordinary matrix multiplication.This forms a group, because the product of two invertible matrices is again invertible, and the inverse of an invertible matrix is invertible, with the identity matrix as the identity element of the group.
Indeed, for PSL(2, p) acts non-trivially on p points if and only if p = 2, 3, 5, 7, or 11; for 2 and 3 the group is not simple, while for 5, 7, and 11, the group is simple – further, it does not act non-trivially on fewer than p points. [note 5] This was first observed by Évariste Galois in his last letter to Chevalier, 1832. [7]
In mathematics, the special linear group SL(n, R) of degree n over a commutative ring R is the set of n × n matrices with determinant 1, with the group operations of ordinary matrix multiplication and matrix inversion. This is the normal subgroup of the general linear group given by the kernel of the determinant