[seqfan] Some questions about groups

[David Newman]

There are two questions further down the page.  I'd appreciate any ideas
anyone can give me about them.

Suppose G is a finite group generated by two elements, a and b,and f(n) is
 the number of elements of length n relative to these two elements. (The
 length of an element is the length of the shortest product of a's and b's

 which yields the given element. The inverses of a and b are not used
explicitly.)


 For example if G is the group given by a^3=1, b^3=1, ba=abq, aq=qa, bq=qb,
 then G has 27 elements and f(0)=1, f(1)=2,
 f(2)=4, f(3)=6, f(4)=7, f(5)=5, f(6)=2, f(7)=0...


(Note: if we had been using the inverses of a and b as well as a and
b, we would have
f(0) = 1, f(1) = 4, f(2) = 8, f(3) = 12, f(4) = 2, f(5) = 0, ...  )



 In this example the counting function f rises to a maximum and then
 decreases, that is, it is unimodal.

 Question 1: Can someone give me an example of a finite group, generated by
 two elements, a and b, where the counting function relative to these two
 elements is not unimodal?


(Note: in de la Harpe's book, "Topics in geometric group theory",
there is an example of a group with non-unimodal growth. However, de
la Harpe is using both the generators and their inverses, so that will
not serve as a counterexample to Question 1.)



 Question 2:  Let G be a finite group,  and let G be generated by two

 elements, a and b. Let f count the number of elements of length n,
relative to a and b. Let F(n) be the set of all such functions , f,
which are counting functions for some G with n elements, and let c(n) be

the number of  functions in F(n). What is the sequence
 c(n)?

 For example: c(3)=1, because the only possible counting function is f(0)=1,
 f(1)=2, there being two elements of length 1.
 c(3) is not zero because there is a group for which this function is the
 counting function, namely Z3.

 Another example: c(4)=1, because the only possible counting function is

 f(0)=1,f(1)=2, f(3)=1.  This is attained for Z4, and for Z2 x Z2 as well.