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If p is a prime number, then any group with p elements is isomorphic to the simple group Z/pZ. A number n is called a cyclic number if Z/nZ is the only group of order n, which is true exactly when gcd(n, φ(n)) = 1. [13] The sequence of cyclic numbers include all primes, but some are composite such as 15. However, all cyclic numbers are odd ...
n. In modular arithmetic, the integers coprime (relatively prime) to n from the set of n non-negative integers form a group under multiplication modulo n, called the multiplicative group of integers modulo n. Equivalently, the elements of this group can be thought of as the congruence classes, also known as residues modulo n, that are coprime to n.
Common group names: Z n: the cyclic group of order n (the notation C n is also used; it is isomorphic to the additive group of Z / nZ) Dih n: the dihedral group of order 2 n (often the notation D n or D 2n is used) K 4: the Klein four-group of order 4, same as Z2 × Z2 and Dih 2. D 2n: the dihedral group of order 2 n, the same as Dih n ...
In group theory, a subfield of abstract algebra, a cycle graph of a group is an undirected graph that illustrates the various cycles of that group, given a set of generators for the group. Cycle graphs are particularly useful in visualizing the structure of small finite groups. A cycle is the set of powers of a given group element a, where an ...
e. In mathematics, specifically group theory, Cauchy's theorem states that if G is a finite group and p is a prime number dividing the order of G (the number of elements in G), then G contains an element of order p. That is, there is x in G such that p is the smallest positive integer with xp = e, where e is the identity element of G.
A cyclic group Z n is a group all of whose elements are powers of a particular element a where a n = a 0 = e, the identity. A typical realization of this group is as the complex n th roots of unity. Sending a to a primitive root of unity gives an isomorphism between the two. This can be done with any finite cyclic group.
For n > 3, except for n = 6, the automorphism group of A n is the symmetric group S n, with inner automorphism group A n and outer automorphism group Z 2; the outer automorphism comes from conjugation by an odd permutation. For n = 1 and 2, the automorphism group is trivial.
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]