# [seqfan] Some questions about groups

David Newman davidsnewman at gmail.com
Thu Feb 11 17:01:03 CET 2010

```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.

```