Exercise · ZW-QUOTIENT-GROUP-007 · ★★☆ Intermediate
Let \(G\) be abelian and \(N\) a subgroup of \(G\). What is \(G/N\)?
Choose one option.
Show answer
- D. Abelian
Explanation
Since \((aN)(bN)=(ab)N=(ba)N=(bN)(aN)\), the quotient is abelian. It need not be cyclic: \((\mathbb{Z}_2\times\mathbb{Z}_2)/\{e\}\) is not.
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-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?