Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

This problem was given and I didn't give and point any ideas about it in the class in time. It says: if the group $G$ is perfect, then $Z\left(G/Z(G)\right)=\{1\}$, where $Z(G)$ is the centre of $G$. Can someone help me to tackle it well. Thank you!

Edit. A group $G$ is called perfect if $G=[G,G]$.

share|improve this question
    
Start with the definition of the centre (or center) of a group, i.e. those elements which commute with every other element of the group. In particular, why does $G/Z(G)$ make sense?? –  hardmath Nov 28 '12 at 10:38
2  
This result is known as Grun's Lemma. Try to google it. –  user26857 Nov 28 '12 at 10:42
add comment

1 Answer 1

up vote 5 down vote accepted

Note first that $gZ(G) \in Z(G/Z(G))$ if and only if $[g,x] \in Z(G)$ for all $x \in G$. Using this fact you can show that when $gZ(G) \in Z(G/Z(G))$, the map $x \mapsto [g,x]$ is a homomorphism $G \rightarrow Z(G)$. Since $Z(G)$ is Abelian and $G$ is perfect, the kernel of this map is all of $G$. Thus $g \in Z(G)$, which proves the claim.

share|improve this answer
    
+1 Very nicely and simply explained. –  DonAntonio Nov 28 '12 at 12:54
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.