Exercise · ZW-SIMPLE-GROUP-008 · ★★☆ Intermediate
For which subgroup \(N=\langle a\rangle\) of \(\mathbb{Z}_{12}\) is the quotient \(\mathbb{Z}_{12}/N\) simple?
Choose one option.
Show answer
- A. \(\langle 2\rangle\)
Explanation
The quotient \(\mathbb{Z}_{12}/\langle a\rangle\) is cyclic of order \(\gcd(a,12)\). It is simple only when that order is prime: \(a=2\) gives \(\mathbb{Z}_2\); \(a=4\) gives \(\mathbb{Z}_4\), \(a=6\) gives \(\mathbb{Z}_6\) and \(a=0\) gives \(\mathbb{Z}_{12}\).
Review the concept: Simple group
A simple group is a nontrivial group \(G\) whose only normal subgroups \(N\) are the trivial subgroup and \(G\) itself.
\[N\trianglelefteq G\ \Rightarrow\ N=\{e\}\ \text{ or }\ N=G\]
More practice problems: simple group
- ZW-SIMPLE-GROUP-001 Which of the following cyclic groups is simple?
- ZW-SIMPLE-GROUP-002 Which of these groups is simple?
- ZW-SIMPLE-GROUP-003 A group G has order 11. What follows?
- ZW-SIMPLE-GROUP-004 Let G be a simple group and φ:G→ H a homomorphism that is not constant. What is kerφ?
- ZW-SIMPLE-GROUP-005 How many normal subgroups does a simple group have?
- ZW-SIMPLE-GROUP-006 Why is S_3 not simple?
- ZW-SIMPLE-GROUP-007 In A_5, the smallest normal subgroup containing the 3-cycle (1 2 3) has order
- ZW-SIMPLE-GROUP-009 What is the order of the smallest nonabelian simple group?