A cyclic group is what remains when an entire group can be recovered from repeatedly applying one operation to one element. This is the first place where the abstract group axioms become computational: instead of tracking arbitrary products of many elements, every element has the form $g^n$ for a single fixed $g$ and some $n \in \mathbb{Z}$. The theory asks how much structure is forced by that compression.

The motivating failure is simple. A group may have many elements, but a chosen element may only move through a small part of it. In the symmetry group of a square, rotating by $90^\circ$ visits the four rotations, while a reflection visits only two symmetries. The question is not merely whether powers exist; the question is when powers of one element account for the whole group.

[example: Rotations and Reflections of a Square] Let $D_8$ be the eight-element symmetry group of a square, with $r$ the rotation by $90^\circ$ and $s$ a reflection. The subscript records the order of the group; some texts write $D_4$ for this same group and reserve $D_8$ for the symmetry group of an octagon. Since four successive $90^\circ$ rotations make one full turn, the powers of $r$ are \begin{align*} r^0=e,\quad r^1=r,\quad r^2=\text{rotation by }180^\circ,\quad r^3=\text{rotation by }270^\circ,\quad r^4=e. \end{align*} Multiplying by further powers of $r$ only repeats this list, because \begin{align*} r^{k+4}=r^k r^4=r^k e=r^k \end{align*} for every $k \in \mathbb{Z}$. Hence \begin{align*} \langle r\rangle=\{e,r,r^2,r^3\}, \end{align*} so $r$ generates exactly the four rotations of the square. A reflection $s$ satisfies $s^2=e$, because applying the same reflection twice restores every point of the square to its original position. Therefore \begin{align*} s^0=e,\quad s^1=s,\quad s^2=e,\quad s^3=s^2s=es=s, \end{align*} and the powers of $s$ are only $e$ and $s$. More generally, every symmetry of a square is either a rotation $r^i$ or a reflection $r^i s$ with $i \in \{0,1,2,3\}$. The rotations have at most four distinct powers, and each reflection has exactly two distinct powers. Thus no element of $D_8$ has eight distinct powers, so no single element generates all of $D_8$. [/example]

This example shows the central theme of the chapter. Cyclic groups are groups whose global structure is governed by one element, and cyclic subgroups measure the part of a larger group seen by repeatedly using one element. The resulting theory connects [groups](/page/Cambridge%20IA%20Groups), modular arithmetic, quotient groups, and the structure of finitely generated abelian groups.

## Definition

The most efficient way to say that a group is controlled by one element is to require every group element to be an integral power of that element. Negative exponents are needed because groups contain inverses, and the exponent $0$ accounts for the identity.

[definition: Cyclic Group] Let $(G, \cdot)$ be a group. The group $G$ is cyclic if there exists an element $g \in G$ such that \begin{align*} G = \{g^n : n \in \mathbb{Z}\}. \end{align*} [/definition]

The definition asserts the existence of a special element, but calculations require us to name that element and compare different possible choices. This motivates a separate definition for the element that performs the generation, because the same cyclic group may have several such elements and many non-generating elements.

[definition: Generator] Let $G$ be a group and let $g \in G$. The element $g$ is a generator of $G$ if \begin{align*} G = \{g^n : n \in \mathbb{Z}\}. \end{align*} [/definition]

If $g$ is a generator, we write $G = \langle g \rangle$ in multiplicative notation. In additive groups, the same idea is written

\begin{align*} G = \langle a \rangle = \{na : n \in \mathbb{Z}\}. \end{align*}

The next issue is what to do when an element does not generate the whole group: its powers still form a meaningful smaller object, and we need a name for that object.

[definition: Cyclic Subgroup] Let $G$ be a group and let $g \in G$. The cyclic subgroup generated by $g$ is \begin{align*} \langle g \rangle = \{g^n : n \in \mathbb{Z}\}. \end{align*} [/definition]

The notation $\langle g \rangle$ would be misleading if the set of powers were not actually a subgroup: closure under multiplication and inverses is not automatic from the definition alone. There is also a minimality issue to settle, because any subgroup containing $g$ must contain every positive and negative power of $g$.

These two concerns lead to the basic structural question behind the notation: does the power-set construction always produce the smallest subgroup that contains the chosen element? The next result supplies that justification, so that $\langle g \rangle$ can be used as a genuine subgroup rather than only as a suggestive set of powers.

[quotetheorem:8235]

The theorem explains why cyclic subgroups are unavoidable in group theory: every element carries a canonical subgroup with it. To understand the size and shape of that subgroup, the next question is when the sequence of powers $e,g,g^2,g^3,\ldots$ first returns to the identity.

[definition: Order of an Element] Let $G$ be a group and let $g \in G$. The order of $g$, denoted $\operatorname{ord}(g)$, is the least positive integer $n \in \mathbb{N}$ such that $g^n=e$, if such an integer exists. If no such integer exists, then $\operatorname{ord}(g)=\infty$. [/definition]

At this point there are two measurements that look related but have not yet been connected: the period of the sequence $e,g,g^2,\ldots$, and the number of elements actually reached by that sequence. Without a theorem identifying them, the notation $\operatorname{ord}(g)$ would describe only a return time, not the size of the subgroup $\langle g \rangle$. The next result is the bridge that lets element orders detect subgroup sizes inside an arbitrary group.

[quotetheorem:8236]