Exercise · ZW-GROUP-003 · ★★☆ Intermediate
Why is (ℤ, ·) not a group?
Choose one option.
Show answer
- D. 2 has no inverse in ℤ
Explanation
In (ℤ, ·) the identity is 1, but 2 has no inverse: there is no integer n with 2 · n = 1. The inverse axiom is missing, although closure, associativity and identity do hold.
Review the concept: Group
A group is a nonempty set \(G\) together with an associative binary operation, with an identity element and an inverse for every element.
\[(G,\ast)\]
More practice problems: group
- ZW-GROUP-001 Which list contains the axioms of a group?
- ZW-GROUP-002 Which structure is a group?
- ZW-GROUP-004 What is the identity of (ℤ, +)?
- ZW-GROUP-005 What is the inverse of 3 in (ℤ, +)?
- ZW-GROUP-006 Which structure is a group with the indicated operation?
- ZW-GROUP-007 ℕ=\0,1,2,…\ with the usual addition is not a group because...
- ZW-GROUP-008 The identity of a group must...
- ZW-GROUP-009 Which of these conditions is NOT a group axiom?