On Mon, 22 Sep 2003, Dan Hoey wrote:
John Conway writes:
Let me ask for the smallest group in which two squares can multiply to a non-square....
It's nice to see this. I was going to look with Gap, and your analysis helped convince me.
I found two such groups of order 16, but I haven't figured out what their names are. A presentation for the first is
<a,b : a^2 = b^4 = (b a b)^2 = (b^-1 a b a)^2 = 1> .
Its squares are 1, b^2, and (a b)^2, and the product of the last two is not a square.
I don't immediately recognise this group in that presentation. I'll play with it offline to see which one it is.
The other group may be presented <a,b : a^4 = b^4 = a b a^-1 b = 1>.
Its squares are 1, a^2, b^2, and again the product of the two non-identity squares is not a square.
This group is called 4:4 in my system. [In general, A:B denotes the split extension of a group of structure A by one of structure B, and it is usually assumed that it's not the direct product, since we'd call that A x B.]
A much harder and more interesting problem is: what's the largest probability we can attain?
The largest among groups up to order 255 is 5/6, in a 192-element group. Gap calls it SmallGroup([192,1022]), but again I don't know a name for it. The smallest presentation I've found is
<a,b,c : a^3 = b^4 = c^4 = [b,c] = [a,b^2] = [a,c^2] = (a c)^3 = (a b)^3 = [c,a^-1 b^-1 a] = 1>
where [x,y] means x y x^-1 y^-1.
This is edged out by probability 27/32, in SmallGroup([384,572]), which I haven't looked at in detail.
Exactly what are these the probabilities of? I was using the probability that a given triple a,b,ab should be of form square,square,nonsquare. Are yours the conditional probability that ab should be a non-square GIVEN that a,b are squares? [I mention that there is a third natural probability around, namely the probability that a^2.b^2 should be a nonsquare.] Regards, JHC