Exercise · ZW-QUOTIENT-GROUP-004 · ★★☆ Intermediate
In \(\mathbb{Z}_{18}/N\) with \(N=\{0,6,12\}\), what is the smallest \(k>0\) with \(k(1+N)=N\)?
Choose one option.
Show answer
- C. 6
Explanation
We need \(k\cdot1=k\in N\) with \(k>0\) minimal: \(k=6\). Thus \(6(1+N)=6+N=N\), while \(1,\dots,5\) are not in \(N\).
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-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?
- ZW-QUOTIENT-GROUP-009 Let G=ℤ_12 and N=langle6rangle. Which statement is true?