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Topic 1 of 8

What is a Group?

A group is one of the simplest yet most powerful structures in mathematics. Four concise rules — closure, associativity, identity, inverses — determine whether a set with a binary operation qualifies. These rules capture the abstract essence of symmetry, and their consequences fill volumes of mathematics and physics.

Key Concepts

Group

A set

G

with a binary operation

\cdot

satisfying four axioms: closure, associativity, identity, and inverses. Written

(G, \cdot)

. The operation need not be commutative.

Order of a Group

The number of elements

|G|

. Can be finite (e.g.

|S_3| = 6

) or infinite (e.g.

|\mathbb{Z}| = \infty

). Continuous groups like

SU(2)

have uncountably infinite order.

Order of an Element

The smallest positive integer

n

such that

g^n = e

. If no such

n

exists the element has infinite order. In

\mathbb{Z}_n

\text{ord}(k) = n/\gcd(n,k)

Abelian Group

A group where

a \cdot b = b \cdot a

for all

a, b \in G

. All cyclic groups are abelian.

S_n

for

n \geq 3

is non-abelian — the non-commutativity of

SU(2)

and

SU(3)

has deep physical consequences.

Cyclic Group

\mathbb{Z}_n

The integers

\{0, 1, \ldots, n{-}1\}

under addition mod

n

. Generated by the single element

1

. Every finite cyclic group of order

n

is isomorphic to

\mathbb{Z}_n

Key Equations

The Four Group Axioms

\begin{aligned} &\text{(1) Closure: } a, b \in G \Rightarrow a \cdot b \in G \\ &\text{(2) Associativity: } (a \cdot b) \cdot c = a \cdot (b \cdot c) \\ &\text{(3) Identity: } \exists\, e \text{ s.t. } e \cdot a = a \\ &\text{(4) Inverses: } \forall a, \exists\, a^{-1} \text{ s.t. } a \cdot a^{-1} = e \end{aligned}

Any set with an operation satisfying all four axioms is a group.

Order of an Element (in

\mathbb{Z}_n

)

\text{ord}(k) = \frac{n}{\gcd(n,\,k)}

The smallest positive

m

with

m \cdot k \equiv 0 \pmod{n}

Worked Example

Order of an Element in $\mathbb{Z}_{12}$

Problem

Find the order of the element $4$ in the group $\mathbb{Z}_{12}$ (integers mod 12 under addition).

Solution

We need the smallest positive $n$ with $n \cdot 4 \equiv 0 \pmod{12}$ .

Compute: $1\cdot4=4$ , $\;2\cdot4=8$ , $\;3\cdot4=12\equiv 0$ . So $\text{ord}(4)=3$ .

Using the formula: $\text{ord}(4) = 12/\gcd(12,4) = 12/4 = 3$ . ✓

Answer

\text{ord}(4) = 3

\mathbb{Z}_{12}

Practice

Exercises

7 problems

1 of 7

In $\mathbb{Z}_8$ (integers mod 8 under addition), what is the order of the element $2$ ?

Your answer

2 of 7

What is $|S_3|$ , the order of the symmetric group on 3 elements (all permutations of 3 objects)?

Your answer

3 of 7

In $\mathbb{Z}_{15}$ , what is the order of the element $5$ ?

Unlock Exercise 3