Group Definition

Level 1 - Math I (Physics) topic page in Groups.

Principle

A group is a set together with a binary operation that behaves like a reversible algebraic process. The operation must combine two elements of the set to produce another element of the same set, and it must satisfy four axioms: closure, associativity, identity, and inverses.

Groups give a compact way to describe reversible transformations, including rotations, reflections, and changes of coordinates in physics.

Binary operation

\[*:G\times G\to G\]

Notation

\(G\) is the set of elements.
\(*\) is the binary operation on \(G\).
\((G,*)\) means the set \(G\) with operation \(*\).
\(a\), \(b\), and \(c\) are elements of \(G\).
\(e\) is the identity element.
\(a^{-1}\) is the inverse of \(a\) with respect to \(*\).

Method

To check whether \((G,*)\) is a group, test the four axioms in order.

Closure: for every \(a,b\in G\), check that \(a*b\in G\).
Associativity: for every \(a,b,c\in G\), check that grouping does not change the product.
Identity: find one element \(e\in G\) that leaves every element unchanged on both sides.
Inverses: for each \(a\in G\), find an element \(a^{-1}\in G\) that combines with \(a\) to give \(e\) on both sides.

Associativity

\[(a*b)*c=a*(b*c)\]

Identity

\[e*a=a*e=a\]

Inverse

\[a*a^{-1}=a^{-1}*a=e\]

Rules

Closure means the operation never leaves the set.
Associativity is about parentheses, not about switching order.
The identity element must work for every element of the group.
Every element must have an inverse inside the same set.
Commutativity, \(a*b=b*a\), is not required for a group.

Examples

The integers under addition form a group, written \((\mathbb Z,+)\).

Closure holds because adding two integers gives an integer.

Integer closure

\[a,b\in\mathbb Z\quad\Rightarrow\quad a+b\in\mathbb Z\]

Associativity holds because addition of integers satisfies:

Addition associativity

\[(a+b)+c=a+(b+c)\]

The identity is \(0\), because adding \(0\) changes nothing.

Additive identity

\[0+a=a+0=a\]

The inverse of \(a\) is \(-a\), because their sum is \(0\).

Additive inverse

\[a+(-a)=(-a)+a=0\]

Subtraction is not the group operation here. The operation is addition, and inverses are described using addition.

Checks

Check that the operation takes two elements of \(G\) back into \(G\).
Check that one identity element works for every element.
Check inverses using the group operation, not a different operation.
Check associativity separately from commutativity.

Symmetries

Group Axioms