Cyclic groups

The group theory is one of the most fundamental parts of our Number theory course. There is a wide use of groups in different areas of mathematics and Number theory is definitely not an exception. We will focus mainly on cyclic groups and their properties. This article should serve as a summary of some basic definitions, properties, and examples. 

Sources:   

  • Drapal, parts 1.1-1.
  • Wiki post on group theory – you can find links to anything you should know ther
  • Socratica has several videos dealing with group theory, especially this one dealing with cyclic groups, or this one explaining Lagrange’s theorem
  • Numberphile video dealing with the Monster group

Theory:

The notion of a group can be introduced in many ways. You can think of it as of a set G with an associative binary operation *, an inverse operation ^{-1} (for each g\in G there exists h\in G such that g*h=h*g=e), and one particular element e of our set, such that e is an identity element for *, i.e., \forall g\in G;\ g*e=e*g=g. We denote this situation by G(*,^{-1},e). \mathbb{Z}(+,-,0), \mathbb{R}\setminus\{0\}(\cdot, ^{-1},1), or S_n(\circ, ^{-1}, id) are among the most typical representatives of groups.

Let G(*,^{-1},e) be a group. A non-empty subset H\subseteq G is a subgroup, if H(*,^{-1},e) is a group. Note that no mater how complicated group you have, some subgroups, namely G and \{e\}, are always present. These subgroups are called trivial for obvious reasons. Subgroups that are not trivial are called proper subgroups.

A group G(*,^{-1},e) is called cyclic, if there exists an element a\in G such that G=\{a^i|i\in\mathbb{Z}\}. Such a is called a generator of G. \mathbb{Z}(+,-,0) is an example of an infinite cyclic group, because it is generated by the element a=1, or b=-1. Another very important example is of the type \mathbb{Z}_n(+,-,0), where all the operations proceed modulo n.

Let G be a group. The order of a group G (denoted by |G|) is the cardinality of the set G. For example, |\mathbb{Z}_n|=n, |S_n|=n!, and the order of \mathbb{Z} is, of course, infinite. The order of an element g\in G is the smallest n\in\mathbb{N} such that g^n=e. If we cannot find any such n, then g is said to be of an infinite order.

LAGRANGE’S THEOREM: Let H be a subgroup of a finite group G. Then |H| divides |G|.

Let G(*_G,^{{-1}_G},e_G) and H(*_H,^{{-1}_H},e_H) be two groups. A map \varphi:G\to H is called a group homomorphism, if \varphi(x{\cdot}_G y)=\varphi(x){\cdot}_H\varphi(y) for any x,y\in G. Injective homomorphism is called a monomorphism, surjective homomorphism is an epimorphism, and if \varphi is a bijection, then we say that it is an isomorphism.

Properties:

  • Every subgroup of \mathbb{Z}(+,-,0) is equal to n\cdot\mathbb{Z}(+,-,0) for some n\in\mathbb{N}_0. Such n is uniquely determined.
  • Let n>0 and A\subseteq\mathbb{Z}_n, A\neq\emptyset. Then A is a subgroup if and only if A=d\cdot\mathbb{Z}_n=\{0,d,2\cdot d, (r-1)\cdot d\}, where d|n, 1\leq d<n, r=\frac{n}{d}.
  • Every subgroup of a cyclic group is cyclic.
  • Let G be a cyclic group. If it is infinite, then G\cong\mathbb{Z}. If G is finite of order n, then it isomorphic to \mathbb{Z}_n. Moreover, in this case, G contains a subgroup of order r \iff r|n. If r|n, then G contains exactly one subgroup of order r.

Examples:

Example 1:

Let G=\mathbb{Z}_4. For each element g\in G, determine the order of g. Deduce, whether G is cyclic or not.

Solution:

\mathbb{Z}_4 consists of elements 0,1,2, and 3. The identity element 0 is always of order 1. Secondly, 1+1=2, 1+1+1=3, and 1+1+1+1=4\equiv 0 in \mathbb{Z}_4, so 1 is of order 4, and, hence, a generator of G, i.e., G is indeed a cyclic group. Next, 2+2=4\equiv 0, which implies that the order of 2 is 2. Finally, 3+3=6\equiv 2, 3+3+3=9\equiv 1, and 3+3+3+3=12\equiv 0. Therefore, 3 is of order 4, which means that it is another generator.

Example 2:

Consider G=\mathbb{Z}_{10}. What are the subgroups of G?

Solution:

We will use Lagrange’s theorem and the last of the Properties. G is cyclic of order 10, so, the order of any subgroup must divide 10. Thus, potential orders of subgroups are 1, 2, 5, and 10. However, we know that for any divisor d, there exists a unique subgroup of order d. There are two trivial subgroups, namely \{0\} of order 1, and G itself of order 10. Next, by the second property, two remaining subgroups are \{0,5\} of order 2, and \{0,2,4,6,8\} of order 5.

Problems:

Problem 1:

Decide whether \mathbb{Q}(+,-,0), \mathbb{R}(+,-,0), or \mathbb{Z}_3(+,-,0) are cyclic groups or not. If so, find all generators.

Problem 2:

Use Lagrange’s theorem to prove that the order of any element in a group G is a divisor of |G|.

Leave a Reply

Your email address will not be published. Required fields are marked *