Exercise · ZW-ORDER-OF-AN-ELEMENT-010 · ★★★ Advanced
In a group, \(a\) and \(b\) commute, \(a\) has order 4 and \(b\) has order 3. What is the order of \(ab\)?
Choose one option.
Show answer
- B. 12
Explanation
Since they commute, \((ab)^n=a^nb^n\), which equals \(e\) only if \(a^n=b^{-n}\) lies in \(\langle a\rangle\cap\langle b\rangle=\{e\}\) (coprime orders); so both 4 and 3 divide \(n\) and the least is \(12\). Example: in \(\mathbb{Z}_{12}\), \(3+4=7\) has order 12.
Review the concept: Order of an element
The order of an element \(a\) of a group with identity \(e\) is the least integer \(n>0\) with \(a^n=e\); if there is none, \(a\) is said to have infinite order.
\[|a|=\min\{\,n\ge 1 : a^n=e\,\}\]
More practice problems: order of an element
- ZW-ORDER-OF-AN-ELEMENT-001 What is the order of the element 3 in the group (ℤ_12,+)?
- ZW-ORDER-OF-AN-ELEMENT-002 What is the order of 2 in the group (ℤ_7^*,·) of units modulo 7?
- ZW-ORDER-OF-AN-ELEMENT-003 What is the order of the permutation (1\ 2\ 3)(4\ 5) in S_5?
- ZW-ORDER-OF-AN-ELEMENT-004 What is the order of 1 in the group (ℤ,+)?
- ZW-ORDER-OF-AN-ELEMENT-005 What is the order of 5 in (ℤ_15,+)?
- ZW-ORDER-OF-AN-ELEMENT-006 The matrix A=[0 -1; 1 0], a 90^∘ rotation, belongs to the group GL_2(ℝ). What is its order?
- ZW-ORDER-OF-AN-ELEMENT-007 An element a of a group has order 6. What is the order of a^2?
- ZW-ORDER-OF-AN-ELEMENT-008 How many elements of order 2 does the symmetric group S_3 have?