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In abstract algebra, every subgroup of a cyclic group is cyclic. Moreover, for a finite cyclic group of order n, every subgroup's order is a divisor of n, and there is exactly one subgroup for each divisor. [1] [2] This result has been called the fundamental theorem of cyclic groups. [3] [4]
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.
Every element a of a group G generates a cyclic subgroup a . If a is isomorphic to Z / n Z {\displaystyle \mathbb {Z} /n\mathbb {Z} } ( the integers mod n ) for some positive integer n , then n is the smallest positive integer for which a n = e , and n is called the order of a .
The lattice of subgroups of a group is the lattice defined by its subgroups, partially ordered by set inclusion. locally cyclic group A group is locally cyclic if every finitely generated subgroup is cyclic. Every cyclic group is locally cyclic, and every finitely-generated locally cyclic group is cyclic. Every locally cyclic group is abelian.
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.
If additionally the lattice satisfies the ascending chain condition, then the group is cyclic. Groups whose lattice of subgroups is a complemented lattice are called complemented groups (Zacher 1953), and groups whose lattice of subgroups are modular lattices are called Iwasawa groups or modular groups (Iwasawa 1941).
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.
Other named normal subgroups of an arbitrary group include the center of the group (the set of elements that commute with all other elements) and the commutator subgroup [,]. [7] [8] More generally, since conjugation is an isomorphism, any characteristic subgroup is a normal subgroup. [9]