Group theory · Definition

Inverse element

groups

Before you start you need to know

In a group, every element can be undone by another: its inverse. Operating \(a\) with \(a^{-1}\) gives the neutral element \(e\), which lets us solve equations inside the group.

Definition

Given \(a\in G\), an element \(b\in G\) is an inverse of \(a\) if operating in both orders gives the neutral element.

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

Also called: or symmetric element, in additive notation

In a group, the inverse of \(a\) is unique, and the inverse of the inverse returns \(a\): \((a^{-1})^{-1}=a\). Also, \((ab)^{-1}=b^{-1}a^{-1}\), with the order reversed.

In \((\mathbb{Z},+)\), the inverse of \(5\) is \(-5\), because \(5+(-5)=0\), the additive identity. In \((\mathbb{Q}^*,\cdot)\), the inverse of \(3/4\) is \(4/3\).

A common mistake is assuming \((ab)^{-1}=a^{-1}b^{-1}\), which only holds in abelian groups, or checking only one order of the operation.

Think about it

What must an inverse of \(a\) satisfy?

That \(a\ast b=b\ast a=e\), i.e. operating in both orders gives the neutral element.

Exercises 11

  1. What is the inverse of 5 in (ℤ, +)?
  2. What is the inverse of 3/4 under multiplication?
  3. In a group, the inverse of an element is...
  4. Which identity is correct?
  5. If a⁻¹ is the inverse of a, then...
  6. The additive inverse of -5 is...
  7. Which statement correctly describes the inverse of an element of a group?
  8. Which condition guarantees that a square matrix has an inverse?
  9. If \(b\) is the inverse of \(a\), which equality always holds?
  10. In a non-abelian group, the inverse of a still satisfies...
  11. In \(\mathbb{Z}_7\setminus\{0\}\) under multiplication, which \(x\) satisfies \(3\cdot x\cdot 3=1\)?

References

  • I. N. Herstein, Topics in Algebra, 2.ª edición, John Wiley & Sons, 1975.