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A cyclic group is a group which is equal to one of its cyclic subgroups: G = g for some element g, called a generator of G. 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.
Formally, the group G is said to have the above presentation if it is isomorphic to the quotient of a free group on S by the normal subgroup generated by the relations R. As a simple example, the cyclic group of order n has the presentation = , where 1 is the group identity.
There is one subgroup dZ for each integer d (consisting of the multiples of d), and with the exception of the trivial group (generated by d = 0) every such subgroup is itself an infinite cyclic group. Because the infinite cyclic group is a free group on one generator (and the trivial group is a free group on no generators), this result can be ...
The cycle graph of a group is not uniquely determined up to graph isomorphism; nor does it uniquely determine the group up to group isomorphism. That is, the graph obtained depends on the set of generators chosen, and two different groups (with chosen sets of generators) can generate the same cycle graph. [2]
The 5th roots of unity in the complex plane form a group under multiplication. Each non-identity element generates the group. In abstract algebra, a generating set of a group is a subset of the group set such that every element of the group can be expressed as a combination (under the group operation) of finitely many elements of the subset and their inverses.
The dihedral group of order 8 requires two generators, as represented by this cycle diagram.. In algebra, a finitely generated group is a group G that has some finite generating set S so that every element of G can be written as the combination (under the group operation) of finitely many elements of S and of inverses of such elements.
Monogenic semigroup of order 9 and period 6. Numbers are exponents of the generator a; arrows indicate multiplication by a. In mathematics, a monogenic semigroup is a semigroup generated by a single element. [1] Monogenic semigroups are also called cyclic semigroups. [2]
A free group on at least two generators is not boundedly generated (see below). The group SL 2 (Z) is not boundedly generated, since it contains a free subgroup with two generators of index 12. A Gromov-hyperbolic group is boundedly generated if and only if it is virtually cyclic (or elementary), i.e. contains a cyclic subgroup of finite index.