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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 $\mathbb{Z}_2$.

Question 1: Do any interesting groups (e.g. central extensions of sporadic simple groups) occur as centralizers (of elements) in $\widetilde{A}_n$?

Question 2: What are their Schur multipliers?

Motivation (feel free to ignore): Write $X = B \widetilde{A}_n$. In trying to find a relatively concrete answer to this question, 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....

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  • $\begingroup$ The centralizer of a conjugacy class is a normal subgroup and hence is either $\tilde{A_n}$ or central. I guess it's not what you mean but the formulation is awkward. $\endgroup$
    – YCor
    Commented Feb 10, 2015 at 7:03
  • $\begingroup$ @YCor: by the centralizer of a conjugacy class I mean the centralizer of an element in that conjugacy class (which is independent, up to isomorphism, of the choice of such an element, so its isomorphism class is a well-defined invariant of the conjugacy class). Sorry if that was unclear. $\endgroup$
    – Qiaochu Yuan
    Commented Feb 10, 2015 at 7:05
  • $\begingroup$ IMHO the answer to Q1 is no, all these centralisers are boring, and look much the same as these in $A_n$, give or take a central extension of a semidirect product of a bunch of $A_k$ and $S_m$... $\endgroup$
    – Dima Pasechnik
    Commented Feb 10, 2015 at 7:26
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    $\begingroup$ You won't find anything at all interesting by looking at centralizers of elements of odd order. It's rather the opposite of what is asked, but double covers of $A_{n}$ do sometimes occur as involution centralizers in sporadic simple groups, eg the Lyons group Ly has an involution centralizer $\hat{A_{11}}.$ $\endgroup$
    – Geoff Robinson
    Commented Feb 10, 2015 at 11:46
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    $\begingroup$ Concerning the motivation - in an answer to the question "third stable homotopy group of spheres via geometry?" I've mentioned work of Igusa from late 70ies which is related $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Feb 10, 2015 at 18:05

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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 $\mathbb{Z}_2$.

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  • $\begingroup$ What do you mean? If you have $x$ and a lift $x'$, then clearly the centralizer of $x'$ is contained in the inverse image of the centralizer of $x$. But the other inclusion is unclear, could you elaborate? $\endgroup$
    – YCor
    Commented Feb 10, 2015 at 8:25
  • $\begingroup$ But they have index at most $2$ in the inverse image of the centralizer of $x$. $\endgroup$
    – Derek Holt
    Commented Feb 10, 2015 at 9:19
  • $\begingroup$ Ah I see, because $y\mapsto [x,y]$ is a homomorphism from the inverse image of the centralizer of $x$ to the center. $\endgroup$
    – YCor
    Commented Feb 10, 2015 at 9:37

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