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Each group is named by Small Groups library as G o i, where o is the order of the group, and i is the index used to label the group within that order. Common group names: Z n: the cyclic group of order n (the notation C n is also used; it is isomorphic to the additive group of Z/nZ) Dih n: the dihedral group of order 2n (often the notation D n ...
A (Z)-group is a group faithfully represented as a doubly transitive permutation group in which no non-identity element fixes more than two points. A (ZT)-group is a (Z)-group that is of odd degree and not a Frobenius group , that is a Zassenhaus group of odd degree, also known as one of the groups PSL(2,2 k +1 ) or Sz(2 2 k +1 ) , for k any ...
dim/C; C n: group operation is addition N 0 0 abelian C n: n: C ×: nonzero complex numbers with multiplication N 0 Z: abelian C: 1 GL(n,C) general linear group: invertible n×n complex matrices: N 0 Z: For n=1: isomorphic to C ×: M(n,C) n 2: SL(n,C) special linear group: complex matrices with determinant. 1 N 0 0 for n=1 this is a single ...
V is the symmetry group of this cross: flipping it horizontally (a) or vertically (b) or both (ab) leaves it unchanged.A quarter-turn changes it. In two dimensions, the Klein four-group is the symmetry group of a rhombus and of rectangles that are not squares, the four elements being the identity, the vertical reflection, the horizontal reflection, and a 180° rotation.
For any A, B, and C subgroups of a group with A ≤ C (A a subgroup of C) then AB ∩ C = A(B ∩ C); the multiplication here is the product of subgroups.This property has been called the modular property of groups (Aschbacher 2000) or (Dedekind's) modular law (Robinson 1996, Cohn 2000).
Hasse diagram of the lattice of subgroups of Z 2 3. The red squares mark the elements of the subsets as they appear in the Cayley table displayed below. There are Z 2 3 itself, seven Z 2 2, seven Z 2 and the trivial group.
Z2 may refer to: Z2 (computer), a computer created by Konrad Zuse; Z2 (company), video game developer; Z2 Comics, a publisher of graphic novels, the quotient ring of the ring of integers modulo the ideal of even numbers, alternatively denoted by / Z 2, the cyclic group of order 2
For a finite cyclic group G of order n we have G = {e, g, g 2, ... , g n−1}, where e is the identity element and g i = g j whenever i ≡ j (mod n); in particular g n = g 0 = e, and g −1 = g n−1. An abstract group defined by this multiplication is often denoted C n, and we say that G is isomorphic to the standard cyclic group C n.