Exercise · ZW-INVERSE-ELEMENT-010 · ★★☆ Intermediate
In a non-abelian group, the inverse of a still satisfies...
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- C. a∗a⁻¹=a⁻¹∗a=e
Explanation
In a non-abelian group, inverses still satisfy a ∗ a⁻¹ = a⁻¹ ∗ a = e. What is lost is commutativity in general, not the existence of two-sided inverses.
Review the concept: Inverse element
Given \(a\in G\), an element \(b\in G\) is an inverse of \(a\) if operating in both orders gives the neutral element.
\[a\ast a^{-1}=a^{-1}\ast a=e\]
More practice problems: inverse element
- ZW-INVERSE-ELEMENT-001 What is the inverse of 5 in (ℤ, +)?
- ZW-INVERSE-ELEMENT-002 What is the inverse of 3/4 under multiplication?
- ZW-INVERSE-ELEMENT-003 In a group, the inverse of an element is...
- ZW-INVERSE-ELEMENT-004 Which identity is correct?
- ZW-INVERSE-ELEMENT-005 If a⁻¹ is the inverse of a, then...
- ZW-INVERSE-ELEMENT-006 The additive inverse of -5 is...
- ZW-INVERSE-ELEMENT-007 Which statement correctly describes the inverse of an element of a group?
- ZW-INVERSE-ELEMENT-008 Which condition guarantees that a square matrix has an inverse?