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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, g2, ... , gn−1}, where e is the identity element and gi = gj whenever i ≡ j (mod n); in particular gn = g0 = e, and g−1 = gn−1.
t. e. In mathematics, specifically in group theory, the direct product is an operation that takes two groups G and H and constructs a new group, usually denoted G × H. This operation is the group-theoretic analogue of the Cartesian product of sets and is one of several important notions of direct product in mathematics.
By definition, the group is cyclic if and only if it has a generator g (a generating set {g} of size one), that is, the powers ,,, …, give all possible residues modulo n coprime to n (the first () powers , …, give each exactly once).
Subgroups of cyclic groups. 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]
The trivial group is the only group of order one, and the cyclic group C p is the only group of order p. There are exactly two groups of order p 2, both abelian, namely C p 2 and C p × C p. For example, the cyclic group C 4 and the Klein four-group V 4 which is C 2 × C 2 are both 2-groups of order 4.
Cyclic order. The months are a cyclic order. In mathematics, a cyclic order is a way to arrange a set of objects in a circle. [nb] Unlike most structures in order theory, a cyclic order is not modeled as a binary relation, such as " a < b ". One does not say that east is "more clockwise" than west. Instead, a cyclic order is defined as a ...
Subgroup. In group theory, a branch of mathematics, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G. Formally, given a group G under a binary operation ∗, a subset H of G is called a subgroup of G if H also forms a group under the operation ∗.
For finite groups, an equivalent definition is that a solvable group is a group with a composition series all of whose factors are cyclic groups of prime order. This is equivalent because a finite group has finite composition length, and every simple abelian group is cyclic of prime order.