Groups (AQA A-Level Further Maths): Revision Notes

Groups

Introduction to groups

A group is a set of mathematical elements that can be combined using a specific operation, following certain rules called axioms. Groups are fundamental structures in abstract algebra and appear throughout mathematics.

The simplest way to understand groups is through examples. Consider the set of permutations of the numbers 1, 2 and 3. A permutation rearranges these numbers in different orders.

Elements and the identity element

The individual items within a group are called elements.

For the permutations of 1, 2 and 3, there are six possible arrangements. These can be written using notation that shows how each number maps to a new position:

$e = \begin{pmatrix} 1 & 2 & 3 \\ 1 & 2 & 3 \end{pmatrix}, \quad a = \begin{pmatrix} 1 & 2 & 3 \\ 1 & 3 & 2 \end{pmatrix}, \quad b = \begin{pmatrix} 1 & 2 & 3 \\ 2 & 1 & 3 \end{pmatrix}$

$c = \begin{pmatrix} 1 & 2 & 3 \\ 2 & 3 & 1 \end{pmatrix}, \quad d = \begin{pmatrix} 1 & 2 & 3 \\ 3 & 1 & 2 \end{pmatrix}, \quad f = \begin{pmatrix} 1 & 2 & 3 \\ 3 & 2 & 1 \end{pmatrix}$

Definition: The identity element (denoted $e$) is the element that leaves every other element unchanged when combined with it.

In the permutations example, element $e$ maps each number to itself: 1→1, 2→2, 3→3. This means the order remains unchanged, making $e$ the identity element.

Closure property

Definition: A set has the closure property under a given operation if combining any two elements from the set always produces another element that is also in the original set.

For the permutations example, if you apply permutation $b$ followed by permutation $c$, the result is permutation $f$, which is already in our set. This property must hold for all possible combinations.

Cayley tables

A Cayley table is a grid showing the result of combining every possible pair of elements in a group. It provides a complete picture of the group's structure.

The table above shows all combinations for the permutation group. The first permutation (from the top row) is applied first, followed by the second permutation (from the left column). The result appears in the corresponding cell.

From this Cayley table, you can verify that every combination produces an element already in the set, confirming the closure property.

Inverse elements

Definition: An inverse element of element $x$ is another element that, when combined with $x$, produces the identity element $e$.

For example, in the permutation group:

• $a \circ a = e$ (so $a$ is its own inverse)
• $c \circ d = e$ (so $c$ and $d$ are inverses of each other)

Definition: A self-inverse is an element that acts as its own inverse, meaning when combined with itself it produces the identity element.

In the permutations example, elements $a$, $b$, $e$ and $f$ are self-inverses, while $d$ is the inverse of $c$ and vice versa.

Associativity

Definition: An operation is associative if the grouping of operations doesn't affect the result. Mathematically, three elements $a$, $b$, $c$ are associative if:

$a(bc) = (ab)c$

This means that $bc$ followed by $a$ gives the same result as $c$ followed by $ab$.

For the permutations example, consider elements $a$, $b$ and $c$:

• $a(bc) = aa = e$ (since combining $b$ and $c$ first gives $a$, then $a$ with $a$ gives $e$)
• $(ab)c = dc = e$ (since combining $a$ and $b$ first gives $d$, then $d$ with $c$ gives $e$)

Both expressions equal $e$, confirming associativity for these elements. This property holds for all combinations in a group.

Formal definition and group axioms

Definition: A group $(S, \circ)$ is a non-empty set $S$ with a binary operation $\circ$ satisfying four axioms:

Note: A binary operation combines two elements of a set according to a specific rule.

These four conditions (collectively called the group axioms) must all be satisfied for a set to qualify as a group.

Order of a group

Definition: The order of a group equals the number of elements in the group.

For the permutations example at the start, the group has 6 elements $(e, a, b, c, d, f)$, so the order is 6.

For the addition modulo 4 example, the group has 4 elements $(0, 1, 2, 3)$, so the order is 4.

Period of an element

Definition: The period (or order) of a particular element $x$ is the smallest positive integer $n$ such that $x^n = e$, where $e$ is the identity.

In other words, the period tells you how many times you must combine an element with itself to return to the identity.

For the permutations example:

• Element $a$ has period 2 because $a^2 = a \circ a = e$
• Element $c$ has period 3 because $c^3 = c \circ c \circ c = e$ (since $c \circ c = d$ and $d \circ c = e$)

Abelian groups

Definition: An abelian group is a group with the additional property of commutativity, meaning the order in which elements are combined doesn't matter.

Mathematically, for all elements $x$ and $y$ in the group:

$xy = yx$

Not all groups are abelian. In the permutations example, $b \circ a \neq a \circ b$, so this group is not abelian.

However, the addition modulo 4 group is abelian because $1 +_4 3 = 3 +_4 1 = 0$ (and this holds for all pairs).

Dihedral groups

Definition: Dihedral groups (denoted $D_n$, where $n$ is the number of sides) represent all symmetries of a regular polygon with $n$ sides.

These symmetries include:

• Rotations about the centre
• Reflections in lines of symmetry

The order of a dihedral group $D_n$ is $2n$ (there are $n$ rotations and $n$ reflections).

Cyclic groups and generators

Definition: A cyclic group contains only rotational symmetries and can be generated by repeatedly applying a single element.

Definition: A generator is an element that, through repeated application, produces all other elements in the group.