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Binary icosahedral group

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Binary tetrahedral group

In mathematics, the binary tetrahedral group, denoted 2T or , is a certain nonabelian group of order 24. It is an extension of the tetrahedral group T or (2,3,3) of order 12 by a cyclic group of order 2, and is the of the tetrahedral group under the 2:1 covering homomorphism Spin(3) → SO(3) of the special orthogonal group by the spin group. It follows that the binary tetrahedral group is a discrete subgroup of Spin(3) of order 24. The complex reflection group named 3(24)3 by G.C.

Covering groups of the alternating and symmetric groups

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 representations of the alternating and symmetric groups. The covering groups were classified in : for n ≥ 4, the covering groups are 2-fold covers except for the alternating groups of degree 6 and 7 where the covers are 6-fold. For example the binary icosahedral group covers the icosahedral group, an alternating group of degree 5, and the binary tetrahedral group covers the tetrahedral group, an alternating group of degree 4.

Homology sphere

In algebraic topology, a homology sphere is an n-manifold X having the homology groups of an n-sphere, for some integer . That is, and for all other i. Therefore X is a connected space, with one non-zero higher Betti number, namely, . It does not follow that X is simply connected, only that its fundamental group is perfect (see Hurewicz theorem). A rational homology sphere is defined similarly but using homology with rational coefficients.

Binary octahedral group

In mathematics, the binary octahedral group, name as 2O or is a certain nonabelian group of order 48. It is an extension of the chiral octahedral group O or (2,3,4) of order 24 by a cyclic group of order 2, and is the of the octahedral group under the 2:1 covering homomorphism of the special orthogonal group by the spin group. It follows that the binary octahedral group is a discrete subgroup of Spin(3) of order 48.

Hurwitz quaternion

In mathematics, a Hurwitz quaternion (or Hurwitz integer) is a quaternion whose components are either all integers or all half-integers (halves of odd integers; a mixture of integers and half-integers is excluded). The set of all Hurwitz quaternions is That is, either a, b, c, d are all integers, or they are all half-integers. H is closed under quaternion multiplication and addition, which makes it a subring of the ring of all quaternions H. Hurwitz quaternions were introduced by .

Dicyclic group

In group theory, a dicyclic group (notation Dicn or Q4n, ) is a particular kind of non-abelian group of order 4n (n > 1). It is an extension of the cyclic group of order 2 by a cyclic group of order 2n, giving the name di-cyclic. In the notation of exact sequences of groups, this extension can be expressed as: More generally, given any finite abelian group with an order-2 element, one can define a dicyclic group.

Binary cyclic group

In mathematics, the binary cyclic group of the n-gon is the cyclic group of order 2n, , thought of as an extension of the cyclic group by a cyclic group of order 2. Coxeter writes the binary cyclic group with angle-brackets, ⟨n⟩, and the index 2 subgroup as (n) or [n]+. It is the binary polyhedral group corresponding to the cyclic group. In terms of binary polyhedral groups, the binary cyclic group is the preimage of the cyclic group of rotations () under the 2:1 covering homomorphism of the special orthogonal group by the spin group.

Superperfect group

In mathematics, in the realm of group theory, a group is said to be superperfect when its first two homology groups are trivial: H1(G, Z) = H2(G, Z) = 0. This is stronger than a perfect group, which is one whose first homology group vanishes. In more classical terms, a superperfect group is one whose abelianization and Schur multiplier both vanish; abelianization equals the first homology, while the Schur multiplier equals the second homology.

Schur multiplier

In mathematical group theory, the Schur multiplier or Schur multiplicator is the second homology group of a group G. It was introduced by in his work on projective representations. The Schur multiplier of a finite group G is a finite abelian group whose exponent divides the order of G. If a Sylow p-subgroup of G is cyclic for some p, then the order of is not divisible by p. In particular, if all Sylow p-subgroups of G are cyclic, then is trivial.

3-manifold

In mathematics, a 3-manifold is a topological space that locally looks like a three-dimensional Euclidean space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane to a small enough observer, all 3-manifolds look like our universe does to a small enough observer. This is made more precise in the definition below. A topological space is a 3-manifold if it is a second-countable Hausdorff space and if every point in has a neighbourhood that is homeomorphic to Euclidean 3-space.