How Are Cyclic Subgroups and Their Generators Related?

A cyclic subgroup is generated by taking all powers of one group element. The same subgroup can have more than one generator, so an element that belongs to a subgroup need not generate it. The key facts below tell you how to calculate a generated subgroup’s order, list its generators, and count the subgroups of a finite cyclic group.

The order of a generated subgroup

Suppose $G=\langle g\rangle$ is a finite cyclic group of order $n$. Every element of $G$ is a power of $g$. For an integer $k$, the subgroup generated by $g^k$ has order

$$|\langle g^k\rangle|=\frac{n}{\gcd(n,k)}.$$

Here, the greatest common divisor is taken using $|k|$ if $k$ is negative; if $k=0$, the formula gives the trivial subgroup of order 1. The formula follows because $(g^k)^r=e$ exactly when $n$ divides $kr$. The smallest positive such $r$ is $n/\gcd(n,k)$.

In particular, $g^k$ generates all of $G$ exactly when $\gcd(n,k)=1$. Thus a cyclic group of order $n$ has $\varphi(n)$ generators, where $\varphi$ is Euler’s totient function.

Generators of a particular subgroup

If $H$ is a cyclic subgroup of order $d$, it has a generator $h$, and its elements are

$$H=\{e,h,h^2,\ldots,h^{d-1}\}.$$

The element $h^j$ generates $H$ exactly when $\gcd(j,d)=1$. If the gcd is greater than 1, taking powers of $h^j$ cannot reach every element of $H$. Therefore, $H$ has exactly $\varphi(d)$ generators.

For example, let $H=\langle h\rangle$ have order 8. Its generators are $h$, $h^3$, $h^5$, and $h^7$, because those are the powers whose exponents are relatively prime to 8. Powers such as $h^2$ and $h^4$ generate proper subgroups instead.

The unique subgroup of each possible order

A finite cyclic group of order $n$ has exactly one subgroup of order $d$ for each positive divisor $d$ of $n, and no subgroup of any other order. If$G=\langle g\rangle$, the subgroup of order$d$ is

$$\left\langle g^{n/d}\right\rangle.$$

To see why it has order $d$, apply the order formula: since $n/d$ divides $n$, the order of $g^{n/d}$ is $n/(n/d)=d$. For uniqueness, any subgroup of a cyclic group is cyclic. If a subgroup has order $d$, its generator must have order $d$, and the elements of order $d$ in $G$ generate the same subgroup $\langle g^{n/d}\rangle$.

Worked example: a cyclic group of order 12

Let $G=\langle x\rangle$ with $|G|=12$. The subgroup generated by $x^4$ has order

$$|\langle x^4\rangle|=\frac{12}{\gcd(12,4)}=3.$$

It consists of $e$, $x^4$, and $x^8$. Its two generators are $x^4$ and $x^8$: within this group, their exponents 1 and 2 are both relatively prime to 3.

The subgroup of order 4 is $\langle x^3\rangle$. Its elements are $e,x^3,x^6,x^9$, and its generators are $x^3$ and $x^9$, corresponding to exponents 1 and 3, both coprime to 4. The subgroup orders in $G$ are precisely the divisors of 12: 1, 2, 3, 4, 6, and 12.

The generators of the whole group are $x^k$ for exponents $k$ relatively prime to 12. Taking exponents from 0 through 11 gives $x,x^5,x^7,x^{11}$, so $G$ has $\varphi(12)=4$ generators.

Infinite cyclic groups

The same ideas have an additive version. An infinite cyclic group is isomorphic to the integers $\mathbb{Z}$. Its subgroups are $m\mathbb{Z}$ for integers $m\geq 0$. When $m>0$, the subgroup $m\mathbb{Z}$ has two generators, $m$ and $-m$. The zero subgroup has just one generator, 0. Equivalently, in multiplicative notation, an infinite cyclic group has two generators of the whole group: $g$ and $g^{-1}$.

Common mistakes

• Confusing membership with generation: An element can belong to $H$ without generating all of $H$. For instance, $x^6$ belongs to $\langle x^3\rangle$ in the example, but it does not generate that order-4 subgroup.
• Using the wrong exponent test: To check whether $g^k$ generates a group of order $n$, test $\gcd(k,n)=1$. For a subgroup of order $d$ generated by $h$, test $\gcd(j,d)=1$ for $h^j$.
• Assuming different generators give different subgroups: Several elements can generate the same subgroup. In a finite cyclic group, each possible subgroup order gives only one subgroup.