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...
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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...
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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...
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Schur multiplier (category Group theory)
groups. The covering groups of the alternating and symmetric groups are of considerable recent interest. Schur's original motivation for studying the...
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information, see Covering groups of the alternating and symmetric groups. Parrott (1976) characterized the Rudvalis group by the centralizer of a central involution...
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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...
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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...
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property. Free groups first arose in the study of hyperbolic geometry, as examples of Fuchsian groups (discrete groups acting by isometries on the hyperbolic...
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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...
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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...
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Inverse Galois problem (redirect from Inverse problem of Galois theory)
Hilbert showed that all symmetric and alternating groups are represented as Galois groups of polynomials with rational coefficients. The polynomial xn + ax...
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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...
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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...
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matrix groups conceptually similar to permutation groups, and the geometry of the action may be usefully exploited to establish properties of the group G....
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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...
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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...
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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...
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between the underlying groups. Hence, there is an isomorphism between the category of groups and the category of discrete groups. Discrete groups can therefore...
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"Lie Groups. Representation Theory and Symmetric Spaces" (PDF). University of Pennsylvania. Media related to Lie groups at Wikimedia Commons Journal of Lie...
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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...
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Icosahedral symmetry (redirect from Icosahedral group)
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...
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Orthogonal matrix (section Group properties)
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)...
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with integer coefficients and determinant 1. The matrices A and −A are identified. The modular group acts on the upper-half of the complex plane by fractional...
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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)...
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Skew-symmetric graphs are identical to the double covering graphs of bidirected graphs. Skew-symmetric graphs were first introduced under the name of antisymmetrical...
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Lie algebra (redirect from Ideal of a Lie algebra)
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...
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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 ) →...
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spaces. The theory of general Kleinian groups was founded by Felix Klein (1883) and Henri Poincaré (1883), who named them after Felix Klein. The special...
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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...
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SL2(R) (category Lie groups)
there are other covering groups of PSL(2, R) corresponding to all n, as n Z < Z ≅ π1 (PSL(2, R)), which form a lattice of covering groups by divisibility;...
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