[math-fun] Re: [math-fun] non-square products of squares?
John Conway <conway@math.princeton.edu> writes:
Since the two best examples are A(4) = 2^2 : 3 and 2^4 : 3 it would seem that 2^6 : 3 deserves examination. Let me try to do that now (despite the fact that I have to lecture in a few minutes).
... I presume 7/32 beats 5/24, but don't have time to check.
Well, of course it does. It seems that 2^6:3 is SmallGroup(192,1541), and yes, it has a s.s=n density of 5/32. I haven't got a way of forming the semidirect product os something that large yet. And pretty much by luck, scanning the 1090235 groups of order 768 from the highest numbered down, the first one that had a derived subgroup of 2^8 was SmallGroup(768,1085321), and it's got a s.s=n density of 85/384 . Assuming that group is 2^8:3, we've got density of 2(1-2^-k)/9 in 2^k:3 for 0(?),2,4,6,8 . I didn't find anything over 1/6 in groups of size 80 or 448. I couldn't check 320. Dan
On Fri, 26 Sep 2003, Dan Hoey wrote:
John Conway <conway@math.princeton.edu> writes:
... I presume 7/32 beats 5/24, but don't have time to check.
Yes of course it does [...]
Let's just do 2^2n:3. The order spectrum of that is 1 , 2^2n - 1, 2^(2n+1), and the squares are precisely the elements of odd order. The argument I gave shows that the number of "bad" triples is (2^2n - 1) x 2^(2n+1), as against a total number of product triples of 2^4n x 9, making a proportion of (2^2n - 1)/(9 x 2^(2n-1)). JHC
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Dan Hoey -
John Conway