Exercise · ZW-SIMPLE-GROUP-001 · ★☆☆ Basic
Which of the following cyclic groups is simple?
Choose one option.
Show answer
- B. \(\mathbb{Z}_{7}\)
Explanation
\(\mathbb{Z}_n\) has one subgroup for each divisor of \(n\), and all are normal because the group is abelian. Only when \(n\) is prime are the only divisors \(1\) and \(n\); since \(7\) is prime, \(\mathbb{Z}_7\) is simple, whereas \(6\), \(8\) and \(9\) have proper divisors.
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-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-008 For which subgroup N=langle arangle of ℤ_12 is the quotient ℤ_12/N simple?
- ZW-SIMPLE-GROUP-009 What is the order of the smallest nonabelian simple group?