Exercise · ZW-SIMPLE-GROUP-004 · ★☆☆ Basic
Let \(G\) be a simple group and \(\varphi:G\to H\) a homomorphism that is not constant. What is \(\ker\varphi\)?
Choose one option.
Show answer
- D. \(\{e\}\)
Explanation
The kernel is a normal subgroup of \(G\), so it is \(\{e\}\) or \(G\). If it were \(G\), the homomorphism would be constant; hence the kernel is \(\{e\}\) and \(\varphi\) is injective.
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-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?