On the number of polynomial functions on
nilpotent groups of class 2
Ecker Jürgen∗
Abstract
In the case of a nilpotent group of class 2 a certain invariant of the group,
the length defined by S. D. Scott can be used to determine the number of
polynomial functions on the group. Sharp upper and lower bounds for this
invariant are determined. It is shown how the length of a group can be
determined from a set of generating elements and the length of all p-groups
up to order p4 is determined as an application.
1
Introduction
All the groups we are treating here are finite. The centre of a group G will be denoted
by Z(G). For the definitions of nilpotency and commutators read the corresponding
chapters in [Hup67]. We deal with polynomial functions on groups of nilpotency
class 2, which will be described in the next theorem. The general definitions of a
polynomial and a polynomial function can be found in [LN73].
Polynomial functions on a group G with pointwise addition and composition form
a near-ring, the polynomial near-ring on G, denoted by P (G).
∗
This study was kindly supported by the Austrian “Fonds zur Förderung der wissenschaftlichen
Forschung” (Project Nr. P11486-TEC). The author wishes to give his thanks to this organisation
and to the referee of this paper for his most welcome and inspiring remarks and suggestions.
1
Theorem 1 If G is of nilpotency class 2, then every polynomial function can be
written in the form
x → gxk [x, h]
(1)
for some g, h ∈ G, k ∈ N.
This gives rise to a formula for the number of polynomial functions on a group of
nilpotency class 2. It is easy to see that if k is the smallest nonzero natural number
such that there exists an element π ∈ G with
xk = [x, π] ∀x ∈ G,
then
|P (G)| = |G|k
|G|
.
|Z(G)|
(2)
One sees that this k is equal to the length λ(G) of the group G as defined in
[Sco69]. Adapted to our situation this definition says that λ(G) = min l(p), where
the minimum is taken over all polynomial functions p of the form (1) with p(x) = 1
for all x ∈ G and l(p) > 0 is the sum of the exponents of x. In this paper we find
that
λ(G)| exp G
(Proposition 1.1 for R(x) = xexp G ) and that
λ(G × H) = lcm(λ(G), λ(H)),
(3)
so we can restrict ourselves to the case of a nilpotent p-group of class ≤ 2.
2
Bounds for λ(G)
Proposition 1 Let G be a p-group of nilpotency class 2 with exp G = pn and λ(G) =
pm . Then
• m ≤ n.
• If p is equal to 2 then m ≥
n+1
.
2
2
• If p is odd then m ≥ n2 .
n
Proof: Clearly m ≤ n, because xp = 1 = [x, 1] for all x ∈ G. Let k = pm and
suppose that π ∈ G is such that xk = [x, π] for all x ∈ G. For x = π it follows that
π k = 1. For x = xπ we get
(xπ)k = [xπ, π]
k
xk π k [x, π]−( 2 ) = [x, π] (by [Hup67, Kapitel III, Hilfssatz 1.3])
k
xk (xk )−( 2 ) = xk
x
So pn = exp G |
k2 (k−1)
2
k2 (k−1)
2
= 1
=
p2m (pm −1)
.
2
m
If p = 2 then pm − 1 is odd, hence pn |p2m−1
and n ≤ 2m − 1. If p > 2 then p − 1 is even, hence pn |p2m and n ≤ 2m. This
completes the proof.
Corollary 1 If G is nilpotent of class 2 and exp G is equal to 4 then λ(G) = exp G =
4.
Proposition 2 Let G be a nilpotent p-group of class 2, exp G = pn and m ∈ N,
m
π ∈ G such that xp = [x, π] for all x ∈ G. Then the order of a possible π is bounded
by
pn−m ≤ ord π ≤ pm .
m
Proof: We see immediately that π p = [π, π] = 1.
The second inequality can be seen as follows: In a p-group of exponent pn there exists
n−m−1
an element a of order pn . Linearity of the commutator operation gives [a, π p
[a, π]
3
pn−m−1
=a
pn−1
6= 1, so in particular π
pn−m−1
]=
6= 1.
Presentations
Suppose that we have a presentation of a group of nilpotency class 2. Is it possible
to determine the number of polynomial functions on the group (i.e. what is λ(G))
from this presentation?
3
Proposition 3 Let G be a nilpotent group of class 2 and generated by a and b.
Then for fixed k and π the following are equivalent:
k
1. [a, b]( 2 ) = 1 and xk = [x, π] holds for x ∈ {a, b}.
2. xk = [x, π] holds for all x ∈ G.
Proof:
• 1 =⇒ 2: Let G be generated by a and b. From the elementary properties of
the commutator in groups of nilpotency class 2 it follows that if
y = aA1 bB1 . . . aAs bBs , then
Qs
[a, y] = [a, a(
i=1
Qs
Ai ) (
b
i=1
Bi )
Qs
] = [a, b(
i=1
Bi )
] = [a, b]c ,
k
for a suitable number c. In particular the assumptions imply [a, y]( 2 ) = 1.
Let x be a word over {a, b}. Now we use induction on the length of x. If
x = ay, then
k
xk = (ay)k = ak y k [a, y]−( 2 ) = [a, π][y, π] = [ay, π] = [x, π].
For x = by the proof is analoguous.
• not 1 =⇒ not 2: If xk = [x, π] does not hold for x ∈ {a, b} then 2 clearly does
k
not hold. So suppose that xk = [x, π] holds for x ∈ {a, b}, but [a, b]( 2 ) 6= 1.
k
Since (ab)k = ak bk ⇐⇒ [a, b]( 2 ) = 1 (use [Hup67, Kapitel 3, Hilfssatz 1.3)])
it follows that (ab)k 6= ak bk . So 2 does not hold (take x = ab ∈ G).
This result can easily be generalized to arbitrarily many generators:
Corollary 2 Let G = hg1 , . . . , gr i be a nilpotent group of class 2. Then for fixed k
and π the following are equivalent:
k
1. [gi , gj ]( 2 ) = 1 for all 1 ≤ i, j ≤ r and xk = [x, π] holds for x ∈ {g1 , . . . , gr }.
2. xk = [x, π] holds for all x ∈ G.
4
4
Minimal examples of class 2 nilpotent p-groups
In a p-group G of exponent pn , the length λ(G) is always a power of p between p
n(+1)
2
and pn . For any prime p, we will give an example of a p-group of arbitrary large
exponent for which the lower bound is sharp.
2l+1
Proposition 4 For any prime p the group G = ha, b; ap
l
l+1
, bp , [a, b] = ap
i is a
semidirect product of Cp2l+1 with Cpl of order p3l+1 and exponent p2l+1 . (By Cn
we denote the cyclic group of order n.) This group is of nilpotency class 2 and
λ(G) = pl+1 .
Proof: That λ(G) = pl+1 follows from Proposition 3. The rest can easily be verified.
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The size of P (G) for all class 2 nilpotent pgroups of order at most p4
For all nilpotent groups G of class 2 and order pn (1 ≤ n ≤ 4) we list the numbers
λ(G). If not otherwise stated the results are obtained with the help of Proposition
3. From (2) it is clear that the number λ(G) contains the information needed to
compute the number of polynomial functions. If not otherwise stated Proposition 3
is used to find λ(G).
1. For n = 1 and n = 2 the resulting groups are all abelian.
2. n = 3:
• For p = 2 all noncyclic groups have exponent 2 or 4, so λ(G) = exp G.
• For p > 2 there are three abelian and two nonabelian nonisomorphic
groups of order p3 (see [Hup67, Kapitel III, Satz 12.4]). One of these
two has exponent p and hence λ(G) = p. The other one is the group
2
ha, b; ap , bp , [a, b] = ap i, which allows λ(G) = p (take π = b). So λ(G) =
p.
3. n = 4:
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• For p = 2 the group ha, b; a8 , b2 , [a, b] = a4 i allows λ(G) = 4 with π = b.
All other groups of nilpotency class 2 have exponent 4, so λ(G) = exp G
by Corollary 1.
• For p > 3 there are 15 nonisomorphic groups of order p4 (see [Hup67, Satz
12.6]). From the remarks before [Hup67, Definition 14.3, Kapitel III] we
see that the groups (9), (10), (12) and (13) are nilpotent of (maximal)
class 3. So the 6 groups (6), (7), (8), (11), (14) and (15) in the list are
nilpotent of class 2.
3
2
(6) The group ha, b; ap , bp , [a, b] = ap i obviously allows λ(G) ≤ p2 (take
π = b), but not λ(G) = p.
2
2
(7) For the group ha, b; ap , bp , [a, b] = ap i we have λ(G) ≤ exp G = p2 .
That λ(G) 6= p can be seen as follows: Every a occuring in π produces
a power of a in [b, π], so the number of a’s in π must be congruent
to 0 modulo p2 . So [b, π] = 1 6= bp .
2
(8) For the group ha, b, c; ap , bp , cp , [a, b] = c−p , all other commutators are
equal to 1i we have λ(G) ≤ exp G = p2 . λ(G) = p is not possible:
Obviously c is in the center of the group, so [c, π] = 1 for all π ∈ G,
but cp 6= 1.
(11) The direct product of Cp with the nonabelian group of order p3 and
exponent p has again exponent p, so λ(G) = p.
(14) For the group G × Cp , where G is the only group of order p3 , where
λ(G) = p < exp G, so λ(G) = p by (3).
(15) The group ha, b, c, d; ap = d, bp , cp , dp , [a, b] = c−1 , all other commutators are equal to 1i has λ(G) = exp G = p2 . λ(G) = p is not possible:
Since a commutes with a, c and d, we have [a, π] = [a, bi ] = c−i and
ap = d. But c−i 6= d, because both c and d have order p and so
c−i = d would imply |G| < p4 .
• For p = 3 a complete list of all groups can be found in [Hup67, Kapitel III,
remarks before Definition 14.3]. There are 6 groups of nilpotency class 2.
As above we can go through the list and find that only 2 groups of order
6
81 have λ(G) < exp G, namely the direct product of C3 with the group of
order 27 with this property and the group ha, b; a27 , b3 , [a, b] = a9 i, which
obviously has λ(G) ≤ 9 (take π = b), and not λ(G) = 3.
References
[Hup67] B. Huppert. Endliche Gruppen I. Springer, 1967.
[LN73] H. Lausch and W. Nöbauer. Algebra of polynomials. North-Holland, Amsterdam, London; American Elsevier Publishing Company, New York, 1973.
[Sco69] S. D. Scott. The arithmetic of polynomial maps over a group and the
structure of certain permutational polynomial groups. i. Monatshefte für
Mathematik, 73:250–267, 1969.
Authors Address:
Ecker Jürgen
Institut für Mathematik
Johannes Kepler Universität Linz
Altenbergerstr. 69, 4040 Linz
Tel.: 0732/2468-9141
Fax.: 0732/2468-10
email: [email protected]
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