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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.
The lattice of subgroups of the infinite cyclic group can be described in the same way, as the dual of the divisibility lattice of all positive integers. If the infinite cyclic group is represented as the additive group on the integers, then the subgroup generated by d is a subgroup of the subgroup generated by e if and only if e is a divisor ...
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.
Cycle, a set equipped with a cyclic order. Necklace (combinatorics), an equivalence classes of cyclically ordered sequences of symbols modulo certain symmetries; Cyclic (mathematics), a list of mathematics articles with "cyclic" in the title; Cyclic group, a group generated by a single element
Even a once-linear group like Z, when bent into a circle, can be thought of as Z 2 / Z. Rieger (1946, 1947, 1948) showed that this picture is a generic phenomenon. For any ordered group L and any central element z that generates a cofinal subgroup Z of L, the quotient group L / Z is a cyclically ordered group. Moreover, every cyclically ordered ...
Thus group theory and the closely related representation theory have many important applications in physics, chemistry, and materials science. Group theory is also central to public key cryptography. The early history of group theory dates from the 19th century.
In group theory, especially, in geometric group theory, the class of free-by-cyclic groups have been deeply studied as important examples. A group G {\displaystyle G} is said to be free-by-cyclic if it has a free normal subgroup F {\displaystyle F} such that the quotient group G / F {\displaystyle G/F} is cyclic .
The cyclic group C 3 consisting of the rotations by 0°, 120° and 240° acts on the set of the three vertices. In mathematics, a group action of a group G on a set S is a group homomorphism from G to some group (under function composition) of functions from S to itself. It is said that G acts on S.