Mathematical group that can be generated as the set of powers of a single element
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In abstract algebra, a cyclic group or monogenous group is a group, denoted Cn (also frequently n or Zn, not to be confused with the commutative ring of p-adic numbers), that is generated by a single element.[1] That is, it is a set of invertible elements with a single associative binary operation, and it contains an element g such that every other element of the group may be obtained by repeatedly applying the group operation to g or its inverse. Each element can be written as an integer power of g in multiplicative notation, or as an integer multiple of g in additive notation. This element g is called a generator of the group.[1]
Every infinite cyclic group is isomorphic to the additive group of Z, the integers. Every finite cyclic group of order n is isomorphic to the additive group of Z/nZ, the integers modulo n. Every cyclic group is an abelian group (meaning that its group operation is commutative), and every finitely generated abelian group is a direct product of cyclic groups.
Every cyclic group of prime order is a simple group, which cannot be broken down into smaller groups. In the classification of finite simple groups, one of the three infinite classes consists of the cyclic groups of prime order. The cyclic groups of prime order are thus among the building blocks from which all groups can be built.
^ ab"Cyclic group", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
In abstract algebra, a cyclicgroup or monogenous group is a group, denoted Cn (also frequently Z {\displaystyle \mathbb {Z} } n or Zn, not to be confused...
quotient group. Subgroups, quotients, and direct sums of abelian groups are again abelian. The finite simple abelian groups are exactly the cyclicgroups of...
|(\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...
In abstract algebra, every subgroup of a cyclicgroup is cyclic. Moreover, for a finite cyclicgroup of order n, every subgroup's order is a divisor of...
cyclicgroup is a group (G, *) in which every finitely generated subgroup is cyclic. Every cyclicgroup is locally cyclic, and every locally cyclic group...
In group theory, a metacyclic group is an extension of a cyclicgroup by a cyclicgroup. That is, it is a group G for which there is a short exact sequence...
the group structure (the rest of the structure is "factored" out). For example, the cyclicgroup of addition modulo n can be obtained from the group of...
group, or phenyl ring, is a cyclicgroup of atoms with the formula C6H5, and is often represented by the symbol Ph (archaically φ). The phenyl group is...
for H. The group G is cyclic, and so are its subgroups. In general, subgroups of cyclicgroups are also cyclic. S4 is the symmetric group whose elements...
examples of finite groups include cyclicgroups and permutation groups. The study of finite groups has been an integral part of group theory since it arose...
{\displaystyle \langle x\rangle } is the cyclic subgroup of the powers of x {\displaystyle x} , a cyclicgroup, and we say this group is generated by x {\displaystyle...
direct product of the cyclicgroups. In the solvable group, C 4 {\displaystyle \mathbb {C} _{4}} is not a normal subgroup. A group G is called solvable...
binary cyclicgroup of the n-gon is the cyclicgroup of order 2n, C 2 n {\displaystyle C_{2n}} , thought of as an extension of the cyclicgroup C n {\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 cyclicgroup Z...
isomorphic to the cyclicgroup 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...
the group within that order. Common group names: Zn: the cyclicgroup of order n (the notation Cn is also used; it is isomorphic to the additive group of...
Consider the cyclicgroup Z3 = (Z/3Z, +) = ({0, 1, 2}, +) and the group of integers (Z, +). The map h : Z → Z/3Z with h(u) = u mod 3 is a group homomorphism...
Corollary — Given a finite group G and a prime number p dividing the order of G, then there exists an element (and thus a cyclic subgroup generated by this...
known ketose is fructose; it mostly exists as a cyclic hemiketal, which masks the ketone functional group. Fatty acid synthesis proceeds via ketones. Acetoacetate...
the class of Coxeter groups. D1 is isomorphic to Z2, the cyclicgroup of order 2. D2 is isomorphic to K4, the Klein four-group. D1 and D2 are exceptional...
easy) group operation. Most cryptographic schemes use groups in some way. In particular Diffie–Hellman key exchange uses finite cyclicgroups. So the...
extension of the cyclicgroup of order 2 by a cyclicgroup of order 2n, giving the name di-cyclic. In the notation of exact sequences of groups, this extension...
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