In abstract algebra, a cyclic group or monogenous group is a group, denoted Cn (also frequently Z {\displaystyle \mathbb {Z} } n or Zn, not to be confused...

cyclic group is a group (G, *) in which every finitely generated subgroup is cyclic. Every cyclic group is locally cyclic, and every locally cyclic group...

quotient group. Subgroups, quotients, and direct sums of abelian groups are again abelian. The finite simple abelian groups are exactly the cyclic groups of...

a} are distinct; despite the name "cyclic group", the powers of the elements do not cycle. An infinite cyclic group is isomorphic to ⁠ ( Z , + ) {\displaystyle...

|(\mathbb {Z} /n\mathbb {Z} )^{\times }|=\varphi (n).} For prime n the group is cyclic, and in general the structure is easy to describe, but no simple general...

a primary cyclic group is a group that is both a cyclic group and a p-primary group for some prime number p. That is, it is a cyclic group of order pm...

binary cyclic group of the n-gon is the cyclic group of order 2n, C 2 n {\displaystyle C_{2n}} , thought of as an extension of the cyclic group C n {\displaystyle...

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...

examples of finite groups include cyclic groups and permutation groups. The study of finite groups has been an integral part of group theory since it arose...

four-group, with four elements, is the smallest group that is not cyclic. Up to isomorphism, there is only one other group of order four: the cyclic group...

the order of S5), because the only group of order 15 is the cyclic group. The largest possible order of a cyclic subgroup (equivalently, the largest...

the group structure (the rest of the structure is "factored out"). For example, the cyclic group of addition modulo n can be obtained from the group of...

{\displaystyle \langle x\rangle } is the cyclic subgroup of the powers of x {\displaystyle x} , a cyclic group, and we say this group is generated by x {\displaystyle...

for groups of the same size. For example, three groups of size 120 are the symmetric group S5, the icosahedral group A5 × Z / 2Z and the cyclic group Z...

for H. The group G is cyclic, and so are its subgroups. In general, subgroups of cyclic groups are also cyclic. S4 is the symmetric group whose elements...

extension of the cyclic group of order 2 by a cyclic group of order 2n, giving the name di-cyclic. In the notation of exact sequences of groups, this extension...

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...

example, the cyclic group C4 and the Klein four-group V4 are both 2-groups of order 4, but they are not isomorphic. Nor need a p-group be abelian; the...

easy) group operation. Most cryptographic schemes use groups in some way. In particular Diffie–Hellman key exchange uses finite cyclic groups. So the...

is cyclic; this implies that its Sylow subgroups are cyclic or generalized quaternion groups. Any group such that all Sylow subgroups are cyclic is called...

the group within that order. Common group names: Zn: the cyclic group of order n (the notation Cn is also used; it is isomorphic to the additive group of...

Galois group. These groups include all of degree no greater than 5. There also are groups known not to have generic polynomials, such as the cyclic group of...

So, finitely generated abelian groups can be thought of as a generalization of cyclic groups. Every finite abelian group is finitely generated. The finitely...

group, or phenyl ring, is a cyclic group of atoms with the formula C6H5, and is often represented by the symbol Ph (archaically φ). The phenyl group is...

and in particular in group theory, a cyclic permutation is a permutation consisting of a single cycle. In some cases, cyclic permutations are referred...

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...

direct product of the cyclic groups. In the solvable group, C 4 {\displaystyle \mathbb {C} _{4}} is not a normal subgroup. A group G is called solvable...

the characters of the circle group. The character group of T {\displaystyle \mathbb {T} } is clearly an infinite cyclic group generated by ⁠ ϕ 1 {\displaystyle...

completed in 2004, is a major milestone in the history of mathematics. The cyclic group G = ( Z / 3 Z , + ) = Z 3 {\displaystyle G=(\mathbb {Z} /3\mathbb {Z}...

isomorphic to the cyclic group Z3, and A0, A1, and A2 are isomorphic to the trivial group (which is also SL1(q) = PSL1(q) for any q). A5 is the group of isometries...