2-group ordering sequences

[Creighton Dement]

Dear Seqfans, 

I just had a little idea concerning what might return some nice
sequences concerning 2-groups, etc. (apparently, every once in a blue
moon, I am still able to think of something which is only indirectly
connected with the floretions). 

Let G be a 2-group (or other group which fits similar requirements -
surely there are others) and let ^ be the operation described at
http://www.crowdog.de/16401.html and let 1 be the unit. 
Note: that page is was written well over a year ago and, if I remember
correctly, I still needed to go over it again to correct some mistakes
(I think these had mostly to do with notation, but who knows...). 

Then there are at least two separate directions which lead to integer
sequences. One direction is finding m(G) (a natural number which depends
on G) such that for all x_1, ..., x_m in G, (x_1,...,x_m)^n orders
itself to (1,...,1) for some n.   I presume this number will be the same
for a large number of groups- it may even provide some sort of
classification scheme for them. The 2nd direction involves recording
what the greatest possible n is corresponding to each m(G). It would be
great if  "theory alone" led to these numbers (I doubt that will be
easy)- until then, an exhaustive search should work to calculate at
least the first few terms. 
 
Sincerely, 
Creighton