Exercise · ZW-SIMPLE-GROUP-007 · ★★☆ Intermediate
In \(A_5\), the smallest normal subgroup containing the \(3\)-cycle \((1\,2\,3)\) has order
Choose one option.
Show answer
- D. 60
Explanation
Since \(A_5\) is simple and the normal subgroup generated by \((1\,2\,3)\) is nontrivial, it must be all of \(A_5\), of order \(60\). It cannot have order \(3\) or \(12\) because it contains every conjugate of the \(3\)-cycle in \(A_5\).
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-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?