Cyclic Group Explorer — Leandro Junes

Group and element

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The additive group. $\mathbb{Z}_n$ is the set $\{0,1,\dots,n-1\}$ with addition modulo $n$ . It is always cyclic. It is not the multiplicative group of units $(\mathbb{Z}/n\mathbb{Z})^\times$ , which need not be.

Group size

n

Element

a

Slide through the elements

The element and its subgroup

Repeated addition

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Show all addition orbits

Multiples of a

k	k·a	k·a mod n

Generators of the group

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An element $a$ generates $\mathbb{Z}_n$ exactly when $\gcd(a,n)=1$ . The number of generators is Euler’s totient $\varphi(n)$ .

The generators

Generators of ⟨a⟩

Distribution of element orders

The order of every element divides $n$ , and for each divisor $d$ there are exactly $\varphi(d)$ elements of order $d$ : the generators of the one subgroup of order $d$ . Hence $\sum_{d\mid n}\varphi(d)=n$ .

order d	elements φ(d)	which ones

All subgroups

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For every divisor $d$ of $n$ there is exactly one subgroup of order $d$ , namely $H_d=\langle n/d\rangle$ ; so the number of subgroups is $\tau(n)$ , the number of divisors. $H_d\subseteq H_e$ exactly when $d\mid e$ .

Subgroup lattice

List of subgroups

order	subgroup	index	generators	elements

Intersection and sum

A second element

b

Quotient by ⟨a⟩

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coset	order in the quotient	elements

About this explorer

The group. $\mathbb{Z}_n$ consists of the remainders $0,1,\dots,n-1$ , added modulo $n$ . The notation is additive throughout: $ka$ means $a$ added to itself $k$ times, $\langle a\rangle$ is the set of all multiples of $a$ , and the order of $a$ is the smallest positive $k$ with $ka\equiv 0 \pmod n$ .

One number decides everything. With $g=\gcd(a,n)$ : the multiples of $a$ are exactly the multiples of $g$ , so $\langle a\rangle=\{0,g,2g,\dots,n-g\}$ and $\operatorname{ord}(a)=n/g$ . For $a=0$ the convention $\gcd(0,n)=n$ gives the trivial subgroup.

Subgroups. Every subgroup of a cyclic group is cyclic, and $\mathbb{Z}_n$ has exactly one of each order dividing $n$ . The lattice drawn on the Subgroups tab is therefore a picture of the divisors of $n$ ordered by divisibility. The intersection of the subgroups of orders $d$ and $e$ has order $\gcd(d,e)$ , and their sum has order $\operatorname{lcm}(d,e)$ .

Quotients and orbits. The cosets of $\langle a\rangle$ are the residue classes modulo $g$ , so $\mathbb{Z}_n/\langle a\rangle\cong\mathbb{Z}_g$ . They are also the orbits of the map $x\mapsto x+a$ : there are $g$ of them, each of length $n/g$ , and the one through 0 is the subgroup itself.

Limits. Everything is computed exactly in your browser for $n$ up to 10¹². The circle is drawn for $n\le 36$ and the lattice for up to 48 subgroups; long lists are shortened and say so.