Exercise · ZW-IDENTITY-ELEMENT-005 · ★★★ Advanced
If e and f are identities of the same group, then...
Choose one option.
Show answer
- B. e = f
Explanation
If e and f are identities: e = e ∗ f (because f is an identity) and e ∗ f = f (because e is an identity). Then e = f. That is why the identity is unique.
Review the concept: Identity element
An element \(e\in G\) is an identity, or neutral element, if it leaves every element of the group unchanged when operating with it.
\[e\ast a=a\ast e=a \quad \forall a\in G\]
More practice problems: identity element
- ZW-IDENTITY-ELEMENT-001 What is the neutral element of (ℤ, +)?
- ZW-IDENTITY-ELEMENT-002 In a group, the identity element is...
- ZW-IDENTITY-ELEMENT-003 The property of the neutral element e is expressed as...
- ZW-IDENTITY-ELEMENT-004 What is the neutral element of (ℚ∖{0}, ·)?
- ZW-IDENTITY-ELEMENT-006 The identity of a group is...
- ZW-IDENTITY-ELEMENT-007 In (ℤ,+), the identity element is...
- ZW-IDENTITY-ELEMENT-008 If e is the identity, what happens with a∗e?
- ZW-IDENTITY-ELEMENT-009 Which expression describes an identity e?