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There's a lot of fuss in certain subfields of algebraic topology about giving fancy interpretations to the rings coming from the cohomology of groups, where "cohomology" is allowed to be taken to be extremely exotic. Some examples of this sort include:

  • the Atiyah-Segal completion theorem, connecting the genuine-equivariant $K$-theory of a point $K_G(\operatorname{pt})$ to the Borel equivariant $K$-theory of a point $K(BG)$,
  • the computation connecting $E^0 B\mathbb{Z}/p^j$ to the $p^j$-torsion of a certain formal group, for suitable cohomology theories $E$, and
  • Neil Strickland's theorem connecting the cohomology of the symmetric groups $E^0 B\Sigma_n$ to the scheme of subgroups of the same above formal group, again for (even more restrictive) suitable cohomology theories $E$.

Recently, Nat Stapleton has extended this list by providing interpretations of groups $G$ which arise in the following manner: fix an integer $n$ and a prime $p$, and consider a collection of $t$ elements $\{\sigma_i \in \Sigma_{p^n}\}_{i=1}^t$ which all pairwise-commute and which are all $p^k$ torsion for some $k \gg 0$. Then, we take $G$ to be the centralizer in the parent group of the subgroup they generate: $$G = C_{\Sigma_{p^n}}(\sigma_1, \ldots, \sigma_t).$$

You can produce a lot of new finite groups using this method (for instance, $D_8$ shows up as such a centralizer in $\Sigma_4$) but I'll bet you can't produce all of them. I'd like to know what's still unknown, and hence my question is:

What is an example of a finite group which cannot be constructed as a direct product of cyclic groups, symmetric groups, and centralizers of the above form? (How do you see that this is the case?)

As a forewarning, I have no idea about the difficulty of this question --- the only finite group theory I know is what I managed to retain from one semester of an undergraduate class. Hopefully it turns out that there's an easy family of examples. :)

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    $\begingroup$ The centralizer of a set $S$ will be the intersection of centralizers of each particular member of $S$ (in any group), and the centralizer of a single element (in a symmetric group) is an internal direct product of internal wreath products of cyclic and symmetric subgroups. Intersecting them sounds like quite a messy business, unfortunately. We could perhaps select generating sets for the cyclic / symmetric subgroups in each centralizer, then investigate interdependence between them (across centralizers). $\endgroup$
    – anon
    Mar 24, 2013 at 19:15
  • $\begingroup$ Maybe you could start by looking at which groups can be constructed as such a centralizer, to rule out certain groups. I don't know if this is useful, but basically these are groups $G = C_{\Sigma_{p^n}}(H)$, where $H$ is an abelian $p$-group. $\endgroup$
    – spin
    Mar 24, 2013 at 21:25
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    $\begingroup$ What about groups with trivial center? $\endgroup$
    – user641
    Mar 25, 2013 at 2:59

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