Group Axioms

Principle

The group axioms are a checklist for proving that an operation is consistent enough to support reversible algebra. They ensure that products stay inside the set, parentheses can be moved safely, there is a do-nothing element, and every element can be undone.

In physics, these same checks appear when transformations are used to describe reversible changes of a system.

Notation

• \(G\) is the set being tested.
• \(*\) is the operation on \(G\).
• \(a\), \(b\), and \(c\) are elements of \(G\).
• \(e\) is the identity element.
• \(a^{-1}\) is the inverse of \(a\).

Method

Test the axioms separately.

1. Closure: show that \(a*b\) lies in \(G\) whenever \(a\) and \(b\) lie in \(G\).
2. Associativity: show that \((a*b)*c=a*(b*c)\) for all choices of \(a\), \(b\), and \(c\) in \(G\).
3. Identity: find a single \(e\in G\) such that \(e*a=a*e=a\) for every \(a\in G\).
4. Inverses: for each \(a\in G\), find \(a^{-1}\in G\) such that \(a*a^{-1}=a^{-1}*a=e\).

Each axiom needs evidence for all elements under discussion. One example can illustrate an axiom, but one example does not prove an axiom for an infinite set.

Rules

• If one group axiom fails, the structure is not a group.
• Closure must be checked before using the result as a group element.
• Associativity is part of the structure; it cannot be replaced by commutativity.
• The identity element must be the same element for every group element.
• Inverses must belong to the same set \(G\).

Examples

Test \(G=\{1,-1\}\) under multiplication.

Closure: every product of two elements in \(G\) is still in \(G\).

The other products are also in the set: \(1\cdot1=1\), \(1\cdot(-1)=-1\), and \((-1)\cdot1=-1\).

Associativity is inherited from real multiplication, so for any \(a,b,c\in G\):

The identity is \(1\), because multiplying by \(1\) leaves each element unchanged.

Each element has an inverse in the set.

Therefore \(\{1,-1\}\) is a group under multiplication.

Checks

• Check closure first; a product outside the set is enough to fail the group test.
• Do not use a different identity for different elements.
• Check both sides when inverses or identities might be one-sided.
• Do not check commutativity in place of associativity.