# Centralizers in the universal central extensions of the alternating groups?

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....

• 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. – YCor Feb 10 '15 at 7:03
• @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. – Qiaochu Yuan Feb 10 '15 at 7:05
• 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 $S_m$... – Dima Pasechnik Feb 10 '15 at 7:26
• 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 $\hat{A_{11}}.$ – Geoff Robinson Feb 10 '15 at 11:46
• 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 – მამუკა ჯიბლაძე Feb 10 '15 at 18:05

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 $\mathbb{Z}_2$.

• 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? – YCor Feb 10 '15 at 8:25
• But they have index at most $2$ in the inverse image of the centralizer of $x$. – Derek Holt Feb 10 '15 at 9:19
• Ah I see, because $y\mapsto [x,y]$ is a homomorphism from the inverse image of the centralizer of $x$ to the center. – YCor Feb 10 '15 at 9:37