Monday, February 8, 2010

Permutation Groups. Automata. Presentations.

I'm in that stage of being interested in something where I want to know waaaaaay more than I know right now. First, look at this problem from Peter Cameron's collection of problems:
Problem 79
The following problem in design theory comes (indirectly) from automata theory.

Let $m$ be an integer greater than 3. Does there exist a $t$-$(2m,m,λ)$ design, for some $t>1$ and some $λ>0$, with the properties

the complement of a block is a block,
the number of blocks properly divides the total number of m-subsets of the point set?
The Hadamard conjecture would imply that the answer is "yes" for all $m>3$; but perhaps there is an easier construction, since $t$ and $λ$ are not specified.
Like, take a look at this pdf.
I emailed Cameron, and he said he's writing a "substantial paper" about the links between permutation groups and Automata. Now, Automata was my first love; the hours I spent on Formal Models homework was the most fun I've had doing homework. But, I know NOTHING yet about permutation groups. I mean, sure I've had classes, but much that I read in Cameron's Permutation Groups book is new to me.

So I end the blog post with a plea to find out what this set is composed of. $G$ is a group, $A$, and $B$ are subgroups, while $\phi$ is an isomomorphism between $A$ and $B$.
$\langle G, t : t^{-1}at=a\phi \text{ for all }a \in A\rangle$
It is larger than $G$, so it could be some sort of product.

I looked up HNN on wiki, and it is supposed to be a presentation. I probably just missed where he defined his notation about presentations. Oh well, back to thinking small; I need to study more. I'm quite quite sick today; my whole body aches something fierce.