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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 ...
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.
These groups are characterized by i) an n-fold proper rotation axis C n; ii) n 2-fold proper rotation axes C 2 normal to C n; iii) a mirror plane σ h normal to C n and containing the C 2 s. The D 1h group is the same as the C 2v group in the pyramidal groups section. The D 8h table reflects the 2007 discovery of errors in older references. [4]
Small groups of prime power order p n are given as follows: Order p: The only group is cyclic. Order p 2: There are just two groups, both abelian. Order p 3: There are three abelian groups, and two non-abelian groups. One of the non-abelian groups is the semidirect product of a normal cyclic subgroup of order p 2 by a cyclic group of order p.
The kth homology group is = / +, the s are the boundary maps of the simplicial complex and the rank of H k is the kth Betti number. Equivalently, one can define it as the vector space dimension of H k ( X ; Q ) since the homology group in this case is a vector space over Q .
The inner automorphism group of D 2 is trivial, whereas for other even values of n, this is D n / Z 2. The cycle graphs of dihedral groups consist of an n-element cycle and n 2-element cycles. The dark vertex in the cycle graphs below of various dihedral groups represents the identity element, and the other vertices are the other elements of ...
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
GF(2) (also denoted , Z/2Z or /) is the finite field with two elements. [1] [a]GF(2) is the field with the smallest possible number of elements, and is unique if the additive identity and the multiplicative identity are denoted respectively 0 and 1, as usual.