Exercise · ZW-SIMPLE-GROUP-010 · ★★★ Advanced
About simple groups, which statement is true?
Choose one option.
Show answer
- B. \(A_5\) is simple and has proper nontrivial subgroups
Explanation
Simple does not mean having no subgroups: \(A_5\) contains, for instance, the subgroup of order \(3\) generated by \((1\,2\,3)\), but none of its proper nontrivial subgroups is normal. The others fail: \(A_5\) is simple and not cyclic, \(S_5\) contains \(A_5\) as a normal subgroup, and \(\mathbb{Z}_p\) is simple with center equal to the whole group.
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-008 For which subgroup N=langle arangle of ℤ_12 is the quotient ℤ_12/N simple?