Exercise · ZW-SIMPLE-GROUP-003 · ★☆☆ Basic
A group \(G\) has order \(11\). What follows?
Choose one option.
Show answer
- A. It is simple, since by Lagrange its only subgroups are \(\{e\}\) and \(G\)
Explanation
The order of a subgroup divides \(11\), which is prime, so the only subgroups are \(\{e\}\) and \(G\), and in particular the only normal ones. Hence \(G\) is simple (and turns out to be cyclic).
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-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?