Exercise · ZW-QUOTIENT-GROUP-009 · ★★★ Advanced
Let \(G=\mathbb{Z}_{12}\) and \(N=\langle6\rangle\). Which statement is true?
Choose one option.
Show answer
- A. \(G/N\) has 6 elements and \(3+N=9+N\)
Explanation
Since \(N=\{0,6\}\), there are \(12/2=6\) cosets. Also \(9-3=6\in N\), so \(3+N=9+N=\{3,9\}\).
Review the concept: Quotient group
Given a normal subgroup \(N\) of a group \(G\), the quotient group \(G/N\) is the set of cosets of \(N\) with the operation induced by multiplying representatives.
\[(aN)(bN)=(ab)N\]
More practice problems: quotient group
- ZW-QUOTIENT-GROUP-001 Let G=ℤ_12 and N=\0,4,8\. How many elements does G/N have?
- ZW-QUOTIENT-GROUP-002 What is the identity element of the quotient group G/N?
- ZW-QUOTIENT-GROUP-003 In ℤ_12/N with N=\0,4,8\, what is (5+N)+(6+N)?
- ZW-QUOTIENT-GROUP-004 In ℤ_18/N with N=\0,6,12\, what is the smallest k>0 with k(1+N)=N?
- ZW-QUOTIENT-GROUP-005 For (aN)(bN)=(ab)N to define an operation on cosets, what must N satisfy?
- ZW-QUOTIENT-GROUP-006 In S_3/A_3, what is big((1 2)A_3big)^2?
- ZW-QUOTIENT-GROUP-007 Let G be abelian and N a subgroup of G. What is G/N?
- ZW-QUOTIENT-GROUP-008 If G/N has a single element, what follows?