Group theory · Definition

Identity element

groups

Before you start you need to know

The identity is the element that leaves any other element unchanged when operated with it. Every group needs one to define inverses.

Definition

An element \(e\in G\) is an identity, or neutral element, if it leaves every element of the group unchanged when operating with it.

\[e\ast a=a\ast e=a \quad \forall a\in G\]

Also called: or neutral element

In a group \(G\) the identity \(e\) is unique and satisfies \(e\ast a=a\ast e=a\) for all \(a\in G\). If there were two identities \(e\) and \(f\), then \(e=e\ast f=f\).

In \((\mathbb{Z},+)\) the identity is \(0\), since \(n+0=0+n=n\) for every integer \(n\). In \((\mathbb{Z},\cdot)\) it is \(1\), since \(n\cdot 1=n\).

A common mistake is to confuse the identity with an inverse: \(a\ast a=e\) says that \(a\) is its own inverse, not that it is neutral. It is also not enough for \(e\ast a=a\) to hold for some \(a\); it must hold for all of them.

Think about it

Why is the identity of a group unique?

Because if there were two identities \(e\) and \(f\), then \(e=e\ast f=f\), so they coincide.

Exercises 10

  1. What is the neutral element of (ℤ, +)?
  2. In a group, the identity element is...
  3. The property of the neutral element e is expressed as...
  4. What is the neutral element of (ℚ∖{0}, ·)?
  5. If e and f are identities of the same group, then...
  6. The identity of a group is...
  7. In (ℤ,+), the identity element is...
  8. If e is the identity, what happens with a∗e?
  9. Which expression describes an identity e?
  10. In (ℤ,·), is 0 an identity?

What comes next

References

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