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...

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...

see Covering groups of the alternating and symmetric groups Outer automorphism group: In general 2. Exceptions: for n = 1, n = 2, it is trivial, and for...

information, see Covering groups of the alternating and symmetric groups. Parrott (1976) characterized the Rudvalis group by the centralizer of a central involution...

groups. The covering groups of the alternating and symmetric groups are of considerable recent interest. Schur's original motivation for studying the...

mathematics, the classification of finite simple groups is a result of group theory stating that every finite simple group is either cyclic, or alternating, or...

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...

property. Free groups first arose in the study of hyperbolic geometry, as examples of Fuchsian groups (discrete groups acting by isometries on the hyperbolic...

in the structure of the Mathieu group M24. The associated extensions SL(n, q) → PSL(n, q) are covering groups of the alternating groups (universal perfect...

considered as the double cover of the symmetric group, 2⋅An → An, with the alternating group being the (rotational) symmetry group of the n-simplex. higher...

matrix groups conceptually similar to permutation groups, and the geometry of the action may be usefully exploited to establish properties of the group G....

For example, three groups of size 120 are the symmetric group S5, the icosahedral group A5 × Z / 2Z and the cyclic group Z / 120Z. The smallest sets on...

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

topological groups are the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity...

between the underlying groups. Hence, there is an isomorphism between the category of groups and the category of discrete groups. Discrete groups can therefore...

under the symmetric group, and Y {\displaystyle Y} is the quotient by the symmetric group of the non-excluded n {\displaystyle n} -tuples. Under the dimension...

subgroups and with an involution whose centralizer is isomorphic to the direct product of the group of order two and the alternating group A5 of order 60...

the circle group, an archetypal example of a Lie group. Lie groups are widely used in many parts of modern mathematics and physics. Lie groups were first...

doubly symmetric a checkerboard pattern of alternatingly a 2-fold rotationally symmetric rectangular tile and its mirror image Examples of group cmm Computer...

The Weyl group of SU(n) is the symmetric group Sn, which is represented by signed permutation matrices (the signs being necessary to ensure that the determinant...

near the identity. They even determine G globally, up to covering spaces. In physics, Lie groups appear as symmetry groups of physical systems, and their...

and for some of their covering groups. These are detailed in Jansen (2005). A further exception in the classification of finite simple groups is the Tits...

being the symmetric group. These can also be related to linear groups over the finite field with five elements, which exhibit the subgroups and covering groups...

isomorphic to the two-element cyclic group Z2. Twofold coverings are characteristic of spin groups. Indeed, in addition to the double coverings Spin+(1, 3)...

covering group SL2(R) → PSL2(R). Further, the modular group has a trivial center, and thus the modular group is isomorphic to the quotient group of B3...

necessary, to work with a covering group of SO(n), the spin group, Spin(n). Likewise, O(n) has covering groups, the pin groups, Pin(n). For n > 2, Spin(n)...

Skew-symmetric graphs are identical to the double covering graphs of bidirected graphs. Skew-symmetric graphs were first introduced under the name of antisymmetrical...

(2005). "The Minimal Degrees of Faithful Representations of the Sporadic Simple Groups and their Covering Groups". LMS Journal of Computation and Mathematics...

understood. The classification of finite simple groups says that most finite simple groups arise as the group G(k) of k-rational points of a simple algebraic...

or (2,3,5) of order 60 by the cyclic group of order 2, and is the preimage of the icosahedral group under the 2:1 covering homomorphism Spin ⁡ ( 3 ) →...