Exercise · ZW-SIMPLE-GROUP-006 · ★★☆ Intermediate
Why is \(S_3\) not simple?
Choose one option.
Show answer
- C. \(A_3\) is a normal subgroup of \(S_3\) of order \(3\)
Explanation
The subgroup \(A_3\) of even permutations has index \(2\), so it is normal; it has order \(3\), so it is neither trivial nor the whole group. Moreover \(S_3\) is nonabelian and its center is trivial.
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-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?