Group theory · Definition
Group
Before you start you need to know
A group is an algebraic structure with a single operation and few axioms. Here you will learn its axioms and how to verify them.
Definition
A group is a nonempty set \(G\) together with an associative binary operation, with an identity element and an inverse for every element.
In a group the identity is unique and each element has a unique inverse. Moreover, the equation \(a x = b\) has the unique solution \(x = a^{-1} b\) in \(G\).
Guiding example: in \((\mathbb{Z},+)\) the identity is \(0\) and the inverse of \(3\) is \(-3\), since \(3+(-3)=0\). This lets us check the axioms one by one.
Typical mistake: assuming every group is commutative. \((\mathbb{Z},+)\) is, but invertible \(2\times 2\) matrices under multiplication are not, so \(ab\) can differ from \(ba\).
Think about it
Why is the identity of a group unique?
If \(e\) and \(e'\) are identities, \(e = e\,e' = e'\), so they coincide.
Exercises 11
- Which list contains the axioms of a group?
- Which structure is a group?
- Why is (ℤ, ·) not a group?
- What is the identity of (ℤ, +)?
- What is the inverse of 3 in (ℤ, +)?
- Which structure is a group with the indicated operation?
- \(\mathbb{N}=\{0,1,2,\ldots\}\) with the usual addition is not a group because...
- The identity of a group must...
- Which of these conditions is NOT a group axiom?
- If an operation has an identity e, what must hold?
- Which statement is correct for \((\mathbb{Q},\ast)\) with \(a\ast b=a+b+1\)?
References
- I. N. Herstein, Topics in Algebra, 2.ª edición, John Wiley & Sons, 1975.