For $n \ge 8$ the Schur multiplier $H\_2(BA\_n, \mathbb{Z})$ (where $A\_n$ denotes the alternating group) stabilizes to $\mathbb{Z}\_2$, and hence there is a universal central extension $\widetilde{A}\_n$ of $A\_n$ by $\mathbb{Z}\_2$.



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> **Question 1:** Do any interesting groups (e.g. central extensions of sporadic simple groups) occur as centralizers (of elements) in $\widetilde{A}\_n$?
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> **Question 2:** What are their Schur multipliers?
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**Motivation** (feel free to ignore): Write $X = B \widetilde{A}\_n$. In trying to find a relatively concrete answer to [this question](https://mathoverflow.net/questions/76541/what-do-the-stable-homotopy-groups-of-spheres-say-about-the-combinatorics-of-fin/), I was led to a map


$$H\_3(X, \mathbb{Z}) \to \pi\_3(\mathbb{S}) \cong \mathbb{Z}\_{24}$$


which I believe is an isomorphism for sufficiently large $n$. In any case, this map, regarded as an element of $H^3(X, \mathbb{Z}\_{24})$, transgresses to a class in


$$H^2(LX, \mathbb{Z}\_{24})$$


where for a group $G$, $LBG$ is the (classifying space of the) adjoint quotient $G/G$, the groupoid whose objects are elements of $g$ and whose morphisms come from conjugation. In particular, this transgressed class restricts to a distinguished family of classes in


$$H^2(C(g), \mathbb{Z}\_{24})$$


where $C(g)$ denotes the centralizer and $g \in \widetilde{A}\_n$. This gives a distinguished family of central extensions of the groups $C(g)$ by $\mathbb{Z}\_{24}$, and it seems interesting to ask what these central extensions are. In particular (feel extremely free to ignore) the Schur multipliers of the sporadic simple groups are all subgroups of $\mathbb{Z}\_{24}$, and it would be nice if this construction in some sense explained that....



Centralizers in the universal central extensions of the alternating groups?
The answer to Q1 is no, all these centralisers are boring, and look much the same as these in $A\_n$ itself. Indeed, think what happens to them under the homomorphism squashing the central $\mathbb{Z}\_2$.


