5. Finite Cyclic Groups

A cyclic group is a group that can be generated by repeatedly applying the group operation to a single element.

A finite cyclic group of order $n$ is just a set of $n$ elements where we "wrap around" after reaching $n$, using addition modulo $n$.

 

Modulo

For any integer $a$ and natural number $n \ge 1$, if you divide $a$ by $n$, you get

$$a = (q \cdot n) + r$$

where $q$ is the quotient and $r$ is the remainder

• The remainder satisfies $0 \le r \le n - 1$
• This remainder is written as

$$a \mod n = r$$

Example

$$23 \mod 4 = 3$$

because $23 = (5 \cdot 4) + 3$

 

Set of Remainders

$$\dfrac{\Z}{n} = \{0, 1, 2, ..., n-1\}$$

these are all the possible remainders when dividing integers by $n$

 

The Operation: Addition Modulo n

We define an operation $\cdot$ on $\dfrac{\Z}{n}$ as

$$a \cdot b = (a + b) \mod n$$

Example

For $n = 5$ $\quad \to \quad 3 \cdot 4$

$$3 + 4 = 7, \quad 7 \mod 5 = 2$$

so $\; 3 \cdot 4 = 2$

 

Finite Cyclic Groups

If a group $G$ has finitely many elements, the order of $G$ is just the number of elements it contains

In $\dfrac{\Z}{n}$ with addition modulo $n$, starting from $0$ and repeatedly adding $1$ generates all the elements

$$0, 1, 2,..., n-1$$

this is known as "clock arithmetic", after reaching $n-1$, you cycle back to $0$

Example

$n=4$, the elements of $\dfrac{\Z}{4}$ are

$$0, \quad 0 + 1 = 1, \quad 1 + 1 = 2, \quad 2 + 1 = 3, \quad 3 + 1 = 0 \mod 4$$

when we get to $4$, $\quad 4 \mod 4 = 0$

so our group $G$ is

$$\{0, 1, 2, 3\}$$

Group Table (Cayley Table)

For small finite groups, we can display all possible compositions $a \cdot b$ in a group table

Procedure

1. List all elements of the group down the left column (choices for $a$)
2. List all elements of the group down the right column (choices for $b$)
3. Fill each cell with the result of $a \cdot b$

Example

Addition modulo $n=4$, $\dfrac{\Z}{4}$