Groups and Subgroups - Key Properties

Cyclic groups are the simplest class of groups, generated by repeated application of a single element. They serve as building blocks for understanding more complex group structures.

DefinitionCyclic Group

A group $G$ is cyclic if there exists an element $g \in G$ such that every element of $G$ can be written as $g^n$ for some integer $n$ . We write $G = \langle g \rangle$ and call $g$ a generator of $G$ .

For additive groups, this means every element is of the form $ng$ for some $n \in \mathbb{Z}$ .

ExampleFundamental Cyclic Groups

Integers: $\mathbb{Z} = \langle 1 \rangle = \langle -1 \rangle$ under addition
Modular arithmetic: $\mathbb{Z}_n = \langle 1 \rangle$ under addition modulo $n$
Roots of unity: The $n$ -th roots of unity $\{\omega^k : k = 0, 1, \ldots, n-1\}$ where $\omega = e^{2\pi i/n}$ , forming the cyclic group $C_n$
Rotation group: Rotations of a regular $n$ -gon by multiples of $2\pi/n$

The structure of cyclic groups is completely determined by their order. Every infinite cyclic group is isomorphic to $\mathbb{Z}$ , and every finite cyclic group of order $n$ is isomorphic to $\mathbb{Z}_n$ .

DefinitionOrder of an Element

The order of an element $g \in G$ , denoted $|g|$ or $\text{ord}(g)$ , is the smallest positive integer $n$ such that $g^n = e$ . If no such $n$ exists, we say $g$ has infinite order.

For a cyclic group $G = \langle g \rangle$ :

If $G$ is finite, then $|G| = |g|$
If $G$ is infinite, then $|g| = \infty$

Remark

For $g \in G$ with finite order $n$ , the cyclic subgroup $\langle g \rangle = \{e, g, g^2, \ldots, g^{n-1}\}$ has exactly $n$ elements. The order of $g$ always divides the order of the group (by Lagrange's theorem).

A key fact about cyclic groups is that all subgroups of a cyclic group are themselves cyclic. If $G = \langle g \rangle$ has order $n$ , then for each divisor $d$ of $n$ , there exists exactly one subgroup of order $d$ , namely $\langle g^{n/d} \rangle$ .

The number of generators of a cyclic group $\mathbb{Z}_n$ equals $\phi(n)$ , Euler's totient function, which counts integers $k$ with $1 \leq k < n$ and $\gcd(k, n) = 1$ . This connects group theory to number theory in a fundamental way.