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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 ...
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)
The manipulations of the Rubik's Cube form the Rubik's Cube group.. In mathematics, a group is a set with an operation that associates an element of the set to every pair of elements of the set (as does every binary operation) and satisfies the following constraints: the operation is associative, it has an identity element, and every element of the set has an inverse element.
Equivalently, the elements of this group can be thought of as the congruence classes, also known as residues modulo n, that are coprime to n. Hence another name is the group of primitive residue classes modulo n. In the theory of rings, a branch of abstract algebra, it is described as the group of units of the ring of integers modulo n.
If the quotient group G/Z(G) is cyclic, G is abelian (and hence G = Z(G), so G/Z(G) is trivial). The center of the Rubik's Cube group consists of two elements – the identity (i.e. the solved state) and the superflip. The center of the Pocket Cube group is trivial. The center of the Megaminx group has order 2, and the center of the Kilominx ...
Some authors define the modular group to be PSL(2, Z), and still others define the modular group to be the larger group SL(2, Z). Some mathematical relations require the consideration of the group GL(2, Z) of matrices with determinant plus or minus one. (SL(2, Z) is a subgroup of this group.) Similarly, PGL(2, Z) is the quotient group GL(2, Z ...
If p is a prime number, then any group with p elements is isomorphic to the simple group Z/pZ. A number n is called a cyclic number if Z/nZ is the only group of order n, which is true exactly when gcd(n, φ(n)) = 1. [13] The sequence of cyclic numbers include all primes, but some are composite such as 15. However, all cyclic numbers are odd ...
In mathematics, the special linear group SL(n, R) of degree n over a commutative ring R is the set of n × n matrices with determinant 1, with the group operations of ordinary matrix multiplication and matrix inversion. This is the normal subgroup of the general linear group given by the kernel of the determinant