# What exactly is a universal central extension?

Let $$G$$ be a group. In An introduction to homological algebra, Chapter 6.9 Weibel defines a universal central extension as a central extension $$0 \to A \to X \to G \to 1,$$ which is initial with respect to all central extensions. In other words, if $$0 \to B \to Y \to G \to 1$$ is any other central extension, then there is a unique homomorphism $$X \to Y$$ which is compatible with the two maps to $$G$$.

The first basic lemma (6.9.2) Weibel proves is that if there exists a universal central extension, then both $$X$$ and $$G$$ are perfect (i.e. $$[G,G] = G$$).

Now both Wikipedia and this notes (Exercise 10) claim that the extension $$0 \to \mathbb Z \to \mathcal B_3 \to \operatorname{PSl}_2(\mathbb Z) \to 1$$ is universal. (For the definition of the extension, see my other question.)

But that apparently contradicts Weibel's lemma, because the group $$\mathcal B_3$$ is not perfect, it's abelization is $$\mathbb Z$$ (by exercise 7 of the above notes). So I guess there is a different notion of universal central extension? This section in Wikipedia seems to also indicate that, but I don't quite understand what they mean, and it seems to mostly care about finite groups? Does that make a difference?

• For those of us who (like me) don't know much about your question, it would be helpful it you could state what your symbols stand for. Like: What kind of algebraic objects are your A, X, G ? Groups? Abelian groups? Something else? Etc. Sep 23 at 0:58
• @DanAsimov Take a look at Wikipedia for the definition of a central extension. My question was aimed at people who already know that, which is why I didn't include it. Sep 24 at 10:53

This question had been discussed at this MO-post. The "usual" definition of a universal central extension $$X$$ of a group $$G$$ assumes that $$G$$ is perfect, and that it is the unique (up to isomorphism) group $$X$$ that is a Schur covering group of $$G$$. In other words, the universal central extension of a perfect group is also perfect - see here.
However, the terminology has been also used as follows: In a universal central extension $$1 \to M \to X \to G \to 1$$ of $$G$$, the image of $$M$$ in $$X$$ is required to be in the commutator subgroup $$[X,X]$$ of $$X$$. In our case, the intersection of the central cyclic subgroup $$\langle x^2 \rangle = \langle y^3 \rangle$$ of $$B_3$$ with $$[B_3,B_3]$$ is trivial.
• Just to verify I understand this correctly: Any central extension $1 \to M \to X \to G \to 1$ with $M \subset [X,X]$ is called universal? So the extension $0 \to c \mathbb Z \to B_3 \to \operatorname{PSl}_2(\mathbb Z) \to 1$ is not universal in this sense, but still called universal for other reasons? May 24 at 12:40
• Yes, some people say that there is no requirement in general for a universal central extension to be perfect - only that its central subgroup $M$ is in its commutator subgroup. May 24 at 12:45
• For a reference see for example here. It is now mostly called a "Schur cover". But in topology it is called "universal central extension". We have a reference already, i.e., the one with Exercise $10$. May 25 at 11:56