What is the order of a cyclic group?

Definition and notation The order of g is the number of elements in ⟨g⟩; that is, the order of an element is equal to the order of its cyclic subgroup. A cyclic group is a group which is equal to one of its cyclic subgroups: G = ⟨g⟩ for some element g, called a generator.

Is a group of order 21 cyclic?

#3 Show that any abelian group of order 21 is cyclic. By Cauchy’s theorem, there are elements x, y of orders 3 and 7, respectively. First, of course, we can note that (x) Π (y) = {e}, since (by Lagrange) the order of the intersection must divide both \(x) = 3 and \(y) = 7, so must be 1.

Is every group of order 23 cyclic?

3 Answers. Here’s an idea, if you can show that the 3, 5, 17, and 23 Sylow subgroups are all normal, then G would be the direct product of these Sylow subgroups. As each of these Sylow subgroups have prime order, they are cyclic. From here, use the Chinese Remainder Theorem to conclude that G is cyclic.

Is group 12 a cyclic order?

Its multiplication table is illustrated above. , 2., 12 are 1, 2, 3, 4, 1, 6, 1, 4, 3, 2, 1, 12. has normal subgroups other than the trivial subgroup and the entire group, it is not a simple group.

How to calculate the Order of a cyclic group?

E.g., the element α= (134)(25) is an element of order 6 in S5 , but α also generates the cyclic subgroup ⟨(134)(25)⟩= {id,α,α2,α3,α4,α5}, whereas (15)(34) generates the cyclic subgroup of order two: {id,(15)(34)}. Remember that all and only those groups which are generated by a single element are cyclic. – amWhy Mar 15 ’17 at 19:24

Why is you ( 12 ) not a cyclic group?

Thus U(12) is not cyclic since none of its elements generate the group. Theorem (4.1 — Criterion for ai= aj). Let G be a group and a 2 G. If |a| = 1, then all distinct powers of a are distinct group elements (ai= aj() i = j). If |a| < 1, say |a| = n, hai = {e,a,a2,…,an 1} and ai= aj() n|ij. Proof. (1) If |a| = 1, 69 n 2 Z+3 an= e.

Is the field extension of the rational numbers cyclic?

The Galois group of the field extension of the rational numbers generated by the n th roots of unity forms a different group, isomorphic to the multiplicative group (Z/nZ) × of order φ(n), which is cyclic for some but not all n (see above). A field extension is called a cyclic extension if its Galois group is cyclic.

What is the subgroup structure of groups of order 24?

This article gives specific information, namely, subgroup structure, about a family of groups, namely: groups of order 24. There are only two prime factors of this number. Order has only two prime factors implies solvable (by Burnside’s -theorem) and hence all groups of this order are solvable groups (specifically, finite solvable groups ).