I suppose that you can pretty easily avoid Burnside's normal p-complement theorem in the case where M is cyclic (as the original poster asked). For if H \subseteq M \cap M^x in this case, then H must be fixed under conjugation by x, as M has a single subgroup of the order of H. Since N_G(H)=M, we get that M is a Frobenius complement.

Benedetti and Lutz have done some experimental work with algorithmically generating discrete Morse collapsing schemes. An announcement of this work can be found at math.kth.se/~brunoben/OWR2012_Benedetti.pdf although as far as I can tell the full paper hasn't yet made it to the arXiv.

@Ryan: I'm reading her questions not as objecting to the use of Euler's formula, but rather objecting to stopping with determining the combinatorial type of the polytopes. (Basically, determining the f-vector.) E.g.: why is the cube the only way to fit 6 4-faces together?

In the edition currently posted on Stanley's website (the July 15 2011 version), this is actually problem 135, rather than 122. It is tagged with [5], which means unsolved.

@Tricia: A major step in the Björner/Kalai paper is that, by applying algebraic shifting, one can reduce problems relating f-vectors and homology down to shifted complexes. Since shifted complexes are (among other nice properties) always bouquets of spheres, the problem you mention indeed looks quite tractable.

I, too, would like to better understand CAT(0) metric spaces. In addition to the Bridson-Haefliger book, I have had recommended to me "A course in metric geometry", by Burago, Burago, and Ivanov.

An idle web search found that the same question came up on math.stackexchange.com a couple of years ago. See math.stackexchange.com/questions/5153/… . In addition to the Usenko result that Bugs gives below, a reference is also given there to a paper of Rosenbaum "Die Untergruppen von halbdirekten Produkten".

I'll also point out that the mathreviews entry for the Thevenaz paper ("Maximal subgroups of direct products") says that the description of subgroups of G x H was earlier in Suzuki's "Group Theory I". On the other hand, the Thevenaz paper is now freely available from any internet connection, while not everyone has Suzuki's book.

Tricia: You're right, I was too hasty. They're somewhat similar, but the distinctness of the $v_j$'s doesn't have to hold in the order complex of the product of posets. (It does seem like there should be some nice interpretation of this in a poset context, but maybe not.)

I'll point out that if $X$ is the order complex of a poset $P$, then $X[m]$ is the order complex of $P \times C_m$, where $C_m$ is the chain with $m$ elements. And there are all sorts of helpful results about product posets...

Following up on JeffE's comment: Ken Brown, in his Cohomology of Groups book (Chapter VII.4), says that the homology version of the nerve lemma "seems to be essentially due to Leray". (Presumably the 1945 paper.) In the exercises he discusses the homotopy version, which he attributes to Weil, 1952.

To expand on Joseph O'Rourke's comment: my favorite way to investigate whether a simplicial complex is a ball:<br><br> 1) Compute the $h$-vector, and see if any entries are negative. (If they are, then it is not Cohen-Macaulay, and in particular not a ball). 2) Search for a shelling. If the complex is shellable, it is quite easy to check if it is a ball. Not every triangulation of a 3-ball is shellable, but many small/nice/naturally occurring such triangulations are. (And if a shelling exists, it is often quick to find.)