# Definition of “succession of central extensions of abelian groups”

What is the meaning of the phrase: "A group $G$ can be realized as a succession of central extensions of abelian groups"?

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Do you know what a central extension is? – Qiaochu Yuan Mar 27 '12 at 2:54

A group $G$ is an extension of $K$ by $H$ (where $K$ and $H$ are groups) if and only if there is a normal subgroup $N\triangleleft G$ such that $N\cong K$ and $G/N\cong H$.

We summarize this by writing $$1 \to K \to G \to H \to 1$$ and saying that the sequence is "exact" (that means that the image of each morphism indicated by an arrow is equal to the kernel of the morphism indicated by the next arrow).

We say that $G$ is a central extension of $K$ by $H$ if and only if there is a central subgroup $N\subseteq Z(G)$ (which is therefore necessarily normal) such that $N\cong K$ and $G/N\cong H$. Necessarily, $K$ must be abelian, but $H$ need not be abelian.

We say that $G$ is a **central extension of abelian groups if $G$ is a central extension of $K$ by $H$, and both $K$ and $H$ are abelian. $G$ may or may not be abelian when it is a central extension of abelian groups. For example, $C_2\otimes C_2$ is a central extension of $C_2$ by $C_2$; and the nonabelian group of order $p^3$ is a central extension of $C_p$ by $C_p\times C_p$ (and is not abelian).

The phrase you suggest is a bit odd, I think; because reading it literally, being a "succession of central extensions of abelian groups" seems to me to be exactly the same as "a central extension of abelian groups". Could you give a citation and context?

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I think the intended meaning is that you start with some collection of abelian groups and repeatedly centrally extend some group in the collection by another group in the collection, then put it back into the collection. – Qiaochu Yuan Mar 27 '12 at 3:31
@Qiaochu: Perhaps; because if $C$ is a central extension of $A$ and $B$, in order to use $C$ in a central extension of abelian groups, then $C$ must be abelian itself, so the "next" extension is simply "a central extension of abelian groups". As opposed to, say, a nilpotent group of class $c$ which can be realized as a succession of $c-1$ central extensions (though only the first one involves two abelian groups). I would still like to see the full context, though... – Arturo Magidin Mar 27 '12 at 3:33