A group is called perfect if we have $[G,G]=G$.
I was wondering in what sense is this group perfect?
I've never really done anything much with perfect groups so I don't really know anything about their properties and so I was wondering in what sense are they perfect? I suppose I see that it's abelianization is trivial but I don't really know what that will give?
Thanks for any help
The word "perfect" is used for groups, fields, numbers and other objects, but in a different sense. A perfect number is a positive integer $n$, which equals the sum of its divisors less than $n$. Euclid named this property "perfect". A perfect group is a group $G$ which has no nontrivial abelian quotients, i.e., with trivial first homology $H_1(G,\mathbb{Z})$. All non-abelian simple groups are perfect, but the converse does not hold, e.g. $SL(2,5)$ is perfect, but not simple. The smallest perfect group is $A_5$. Such groups were first called (in german) "vollkommen", and than later "perfekt". The statement that the quotient of a perfect group by its center is centerless, is called Grün's lemma.