J. Number Th. 133 (2013) 4188–4199

Sylow p-groups of polynomial permutations on the integers mod p^{n}

Sophie Frisch and Daniel Krenn

Abstract. We enumerate and describe the Sylow p-groups of the groups of polynomial permutations of the integers mod p^{n} for n ≥ 1 and of the pro-finite group which is ths projective limit of these groups. MSC 2000: primary 20D20, secondary 11T06, 13M10, 11C08, 13F20, 20E18.

1. Introduction

Fix a prime p and let n ∈ ℕ. Every polynomial f ∈ ℤ[x] defines a function from ℤ_{p^{n}} = ℤ/p^{n}ℤ to itself. If this function happens to be bijective, it is called a polynomial permutation of ℤ_{p^{n}}. The polynomial permutations of ℤ_{p^{n}} form a group (G_{n}, ∘) with respect to composition. The order of this group has been known since at least 1921 (Kempner [10]) to be

|G_{2}| = p!(p − 1)^{p}p^{p} and |G_{n}| = p!(p − 1)^{p}p^{p}p^{∑_{k=3}^{n}β(k)} for n ≥ 3,

where β(k) is the least n such that p^{k} divides n!, but the structure of (G_{n}, ∘) is elusive. (See, however, Nöbauer [15] for some partial results). Since the order of G_{n} is divisible by a high power of (p − 1) for large p, even the number of Sylow p-groups is not obvious.

We will show that there are (p − 1)!(p − 1)^{p−2} Sylow p-groups of G_{n} and describe these Sylow p-groups, see Theorem 5.1 and Corollary 5.2.

Some notation: p is a fixed prime throughout. A function g: ℤ_{p^{n}} → ℤ_{p^{n}} arising from a polynomial in ℤ_{p^{n}}[x] or, equivalently, from a polynomial in ℤ[x], is called a polynomial function on ℤ_{p^{n}}. We denote by (F_{n}, ∘) the monoid with respect to composition of polynomial functions on ℤ_{p^{n}}. By monoid, we mean semigroup with an identity element. Let (G_{n}, ∘) be the group of units of (F_{n}, ∘), which is the group of polynomial permutations of ℤ_{p^{n}}.

Sophie Frisch is supported by the Austrian Science Fund (FWF), grant P23245-N18.

Daniel Krenn is supported by the Austrian Science Fund (FWF), project W1230 doctoral program “Discrete Mathematics”.

SYLOW GROUPS OF GROUPS OF POLYNOMIAL PERMUTATIONS

Since every function induced by a polynomial preserves congruences modulo ideals, there is a natural epimorphism mapping polynomial functions on ℤ_{p^{n+1}} onto polynomial functions on ℤ_{p^{n}}, and we write it as π_{n} : F_{n+1} → F_{n}. If f is a polynomial in ℤ[x] (or in ℤ_{p^{m}}[x] for m ≥ n) we denote the polynomial function on ℤ_{p^{n}}[x] induced by f by [f]_{p^{n}}.

The order of F_{n} and that of G_{n} have been determined by Kempner [10] in a rather complicated manner. His results were cast into a simpler form by Nöbauer [14] and Keller and Olson [9] among others. Since then there have been many generalizations of the order formulas to more general finite rings [16, 13, 2, 6, 1, 8, 7]. Also, polynomial permutations in several variables (permutations of (ℤ_{p^{n}})^{k} defined by k-tuples of polynomials in k variables) have been looked into [5, 4, 19, 17, 18, 11].

2. Polynomial functions and permutations

To put things in context, we recall some well-known facts, to be found, among other places, in [10, 14, 3, 9]. The reader familiar with polynomial functions on finite rings is encouraged to skip to section 3. Note that we do not claim anything in section 2 as new.

Definition. For p prime and n ∈ ℕ, let

α_{p}(n) = ∑_{k=1}^{∞} [\frac{n}{p^{k}} ]  and  β_{p}(n) = min\{m | α_{p}(m) ≥ n\}.

If p is fixed, we just write α(n) and β(n).

Notation. For k ∈ ℕ, let (x)_{k} = x(x − 1) … (x − k + 1) and (x)_{0} = 1. We denote p-adic valuation by v_{p}.

2.1 Fact.

(1) α_{p}(n) = v_{p}(n!).

(2) For 1 ≤ k ≤ p, β_{p}(k) = kp and for k > p, β_{p}(k) < kp.

(3) For all n ∈ ℤ, v_{p}((n)_{k}) ≥ α_{p}(k); and v_{p}((k)_{k}) = v_{p}(k!) = α_{p}(k).

Proof. Easy. □

Remark. The sequence (β_{p}(n))_{n=1}^{∞} is obtained by going through the natural numbers in increasing order and repeating each k ∈ ℕ v_{p}(k) times. For instance, β_{2}(n) for n ≥ 1 is: 2, 4, 4, 6, 8, 8, 8, 10, 12, 12, 14, 16, 16, 16, 16, 18, 20, 20, . . ..

The falling factorials (x)_{0} = 1, (x)_{k} = x(x − 1) … (x − k + 1), k > 0, form a basis of the free ℤ-module ℤ[x], and representation with respect to this basis gives a convenient canonical form for a polynomial representing a given polynomial function on ℤ_{p^{n}}.

2.2 Fact. (cf. Keller and Olson [9]) A polynomial f ∈ ℤ[x], f = ∑_{k} a_{k} (x)_{k}, induces the zero-function mod p^{n} if and only if a_{k} ≡ 0 mod p^{n−α(k)} for all k (or, equivalently, for all k < β(n)).

Proof. Induction on k using the facts that (m)_{k} =0 for m<k, that v_{p}((n)_{k})≥α_{p}(k) for all n ∈ ℤ, and that v_{p}((k)_{k}) = v_{p}(k!) = α_{p}(k). □

2.3 Corollary. (cf. Keller and Olson [9]) Every polynomial function on ℤ_{p^{n}} is represented by a unique f ∈ ℤ[x] of the form f = ∑_{k=0}^{β(n)−1} a_{k} (x)_{k}, with 0 ≤ a_{k} < p^{n−α(k)} for all k.

Comparing the canonical forms of polynomial functions mod p^{n} with those mod p^{n−1} we see that every polynomial function mod p^{n−1} gives rise to p^{β(n)} different polynomial functions mod p^{n}:

2.4 Corollary. (cf. Keller and Olson [9]) Let (F_{n}, ∘) be the monoid of polynomial functions on ℤ_{p^{n}} with respect to composition and π_{n} : F_{n+1} → F_{n} the canonical projection.

(1) For all n ≥ 1 and for each f ∈ F_{n} we have |π_{n}^{−1}(f)| = p^{β(n+1)}.

(2) For all n ≥ 1, the number of polynomial functions on ℤ_{p^{n}} is

|F_{n}| = p^{∑_{k=1}^{n}β(k)}.

Notation. We write [f]_{p^{n}} for the function defined by f ∈ ℤ[x] on ℤ_{p^{n}}.

2.5 Lemma. Every polynomial f ∈ ℤ[x] is uniquely representable as

f(x) = f_{0}(x) + f_{1}(x)(x^{p} − x) + f_{2}(x)(x^{p} − x)^{2} + … + f_{m}(x)(x^{p} − x)^{m} + …

with f_{m} ∈ ℤ[x], deg f_{m} < p, for all m ≥ 0. Now let f, g ∈ ℤ[x].

(1) If n ≤ p, then [f]_{p^{n}} = [g]_{p^{n}} is equivalent to: f_{k} = g_{k} mod p^{n−k}ℤ[x] for 0 ≤ k < n.

(2) [f]_{p^{2}} = [g]_{p^{2}} is equivalent to: f_{0} = g_{0} mod p^{2}ℤ[x] and f_{1} = g_{1} mod pℤ[x].

(3) [f]_{p} = [g]_{p} and [f′]_{p} = [g′]_{p} is equivalent to: f_{0} = g_{0} mod pℤ[x] and f_{1} = g_{1} mod pℤ[x].

Proof. The canonical representation is obtained by repeated division with remainder by (x^{p} −x), and uniqueness follows from uniqueness of quotient and remainder of polynomial division. Note that [f]_{p} = [f_{0}]_{p} and [f′]_{p} = [f′_{0} − f_{1}]_{p}. This gives (3).

Denote by f ∼ g the equivalence relation f_{k} = g_{k} mod p^{n−k}ℤ[x] for 0 ≤ k < n. Then f ∼ g implies [f]_{p^{n}} = [g]_{p^{n}}. There are p^{p+2p+3p+...+np} equivalence classes of ∼ and p^{β(1)+β(2)+β(3)+...+β(n)} different [f]_{p^{n}}. For k ≤ p, β(k) = kp. Therefore the equivalence relations f ∼ g and [f]_{p^{n}} = [g]_{p^{n}} coincide. This gives (1), and (2) is just the special case n = 2. □

We can rephrase this in terms of ideals of ℤ[x].

2.6 Corollary. For every n ∈ ℕ, consider the two ideals of ℤ[x]

I_{n} = {f ∈ ℤ[x] | f(ℤ) ⊆ p^{n}ℤ}  and  J_{n} = ({p^{n−k}(x^{p} − x)^{k} | 0 ≤ k ≤ n}).

Then [ℤ[x] : I_{n}] = p^{β(1)+β(2)+β(3)+...+β(n)} and [ℤ[x] : J_{n}] = p^{p+2p+3p+...+np}. Therefore, J_{n} = I_{n} for n ≤ p, whereas for n > p, J_{n} is properly contained in I_{n}.

Proof. J_{n} ⊆ I_{n}. The index of J_{n} in ℤ[x] is p^{p+2p+3p+...+np}, because f ∈ J_{n} if and only if f_{k} = 0 mod p^{n−k}ℤ[x] for 0 ≤ k < n in the canonical representation of Lemma 2.5. The index of I_{n} in ℤ[x] is p^{β(1)+β(2)+β(3)...+β(n)} by Corollary 2.4 (2) and [ℤ[x] : I_{n}] < [ℤ[x] : J_{n}] if and only if n > p by Fact 2.1 (2). □

2.7 Fact. (cf. McDonald [12]) Let n ≥ 2. The function on ℤ_{p^{n}} induced by a polynomial f ∈ ℤ[x] is a permutation if and only if

(1) f induces a permutation of ℤ_{p} and

(2) the derivative f′ has no zero mod p.

2.8 Lemma. Let [f]_{p^{n}} and [f]_{p} be the functions defined by f ∈ ℤ[x] on ℤ_{p^{n}} and ℤ_{p}, respectively, and [f′]_{p} the function defined by the formal derivative of f on ℤ_{p}. Then

(1) [f]_{p^{2}} determines not just [f]_{p}, but also [f′]_{p}.

(2) Let n ≥ 2. Then [f]_{p^{n}} is a permutation if and only if [f]_{p^{2}} is a permutation.

(3) For every pair of functions (α, β), α: ℤ_{p} → ℤ_{p}, β : ℤ_{p} → ℤ_{p}, there are exactly p^{p} polynomial functions [f]_{p^{2}} on ℤ_{p^{2}} with [f]_{p} = α and [f′]_{p} = β.

(4) For every pair of functions (α, β), α : ℤ_{p} → ℤ_{p} bijective, β : ℤ_{p} → ℤ_{p} \ {0}, there are exactly p^{p} polynomial permutations [f]_{p^{2}} on ℤ_{p^{2}} with [f]_{p} = α and [f′]_{p} = β.

Proof. (1) and (3) follow immediately from Lemma 2.5 for n = 2 and (2) and (4) then follow from Fact 2.7. □

2.9 Remark. Fact 2.7 and Lemma 2.8 (2) imply that

(1) for all n ≥ 1, the image of G_{n+1} under π_{n} : F_{n+1} → F_{n} is contained in G_{n} and (2) for all n ≥ 2, the inverse image of G_{n} under π_{n} : F_{n+1} → F_{n} is G_{n+1}.

We denote by π_{n}: G_{n+1} → G_{n} the restriction of π_{n} to G_{n}. This is the canonical epimorphism from the group of polynomial permutations on ℤ_{p^{n+1}} onto the group of polynomial permutations on ℤ_{p^{n}}.

The above remark allows us to draw conclusions on the projective system of groups G_{n} from the information in Corollary 2.4 concerning the projective system of monoids F_{n}.

2.10 Corollary. Let n ≥ 2, and π_{n} : G_{n+1} → G_{n} the canonical epimorphism from the group of polynomial permutations on ℤ_{p^{n+1}} onto the group of polynomial permutations on ℤ_{p^{n}}. Then

| ker(π_{n})| = p^{β(n+1)}.

2.11 Corollary. (cf. Kempner [10] and Keller and Olson [9]) The number of polynomial permutations on ℤ_{p^{2}} is

|G_{2}| = p!(p − 1)^{p}p^{p},

and for n ≥ 3 the number of polynomial permutations on ℤ_{p^{2}} is

|G_{n}| = p!(p − 1)^{p}p^{p}p^{∑_{k=3}^{n}β(k)}.

Proof. In the canonical representation of f ∈ℤ[x] in Lemma 2.5, there are p!(p−1)^{p} choices of coefficients mod p for f_{0} and f_{1} such that the criteria of Fact 2.7 for a polynomial permutation on ℤ_{p^{2}} are satisfied. And for each such choice there are p^{p} possibilities for the coefficients of f_{0} mod p^{2}. The coefficients of f_{0} mod p^{2} and those of f_{1} mod p then determine the polynomial function mod p^{2}. So |G_{2}| = p!(p − 1)^{p}p^{p}. The formula for |G_{n}| then follows from Corollary 2.10. □

This concludes our review of polynomial functions and polynomial permutations on ℤ_{p^{n}}. We will now introduce a homomorphic image of G_{2} whose Sylow p-groups bijectively correspond to the Sylow p-groups of G_{n} for any n ≥ 2.

3. A group between G_{1} and G_{2}

Into the projective system of monoids (F_{n}, ∘) we insert an extra monoid E between F_{1} and F_{2} by means of monoid epimorphisms θ : F_{2} → E and ψ : E → F_{1} with ψθ = π_{1}.

F_{1} ⟵^{ψ} E ⟵^{θ} F_{2} ⟵^{π_{2}} F_{3} ⟵^{π_{3}} …

The restrictions of θ to G_{2} and of ψ to the group of units H of E will be group-epimorphisms, so that we also insert an extra group H between G_{1} and G_{2} into the projective system of the G_{i}.

G_{1} ⟵^{ψ} H ⟵^{θ} G_{2} ⟵^{π_{2}} G_{3} ⟵^{π_{3}} …

In the following definition of E and H, f and f′ are just two different names for functions. The connection with polynomials and their formal derivatives suggested by the notation will appear when we define θ and ψ.

Definition. We define the semi-group (E, ∘) by

E = {(f, f′) | f : ℤ_{p} → ℤ_{p} f′ : ℤ_{p} → ℤ_{p}}

(where f and f′ are just symbols) with law of composition

(f, f′) ∘ (g, g′) = (f ∘ g, (f′ ∘ g) · g′).

Here (f ∘ g)(x) = f(g(x)) and ((f′ ∘ g) · g′)(x) = f′(g(x)) · g′(x).

We denote by (H, ∘) the group of units of E.

The following facts are easy to verify:

3.1 Lemma.

(1) The identity element of E is (ι, 1), with ι denoting the identity function on ℤ_{p} and 1 the constant function 1.

(2) The group of units of E has the form

H = {(f, f′) | f : ℤ_{p} → ℤ_{p} bijective, f′ : ℤ_{p} → ℤ_{p} \ {0}}.

(3) The inverse of (g, g′) ∈ H is

(g, g′)^{−1} = (g^{−1}, \frac{1}{g′ ∘ g^{−1}} ),

where g^{−1} is the inverse permutation of the permutation g and 1/a stands for the multiplicative inverse of a non-zero element a ∈ ℤ_{p}, such that

(\frac{1}{g′ ∘ g^{−1}} )(x) = \frac{1}{g′(g^{−1}(x))}

means the multiplicative inverse in ℤ_{p} \ {0} of g′(g^{−1}(x)).

Note that H is a semidirect product of (as the normal subgroup) a direct sum of p copies of the cyclic group of order p − 1 and (as the complement acting on it) the symmetric group on p letters, S_{p}, acting on the direct sum by permuting its components. In combinatorics, one would call this a wreath product (designed to act on the left) of the abstract group C_{p−1} by the permutation group S_{p} with its standard action on p letters. (Group theorists, however, have a narrower definition of wreath product, which is not applicable here.)

Now for the homomorphisms θ and ψ.

Definition. We define ψ : E ⟶ F_{1} by ψ(f, f′) = f. As for θ : F_{2} → E, given an element [g]_{p^{2}} ∈ F_{2}, set θ([g]_{p^{2}}) = ([g]_{p}, [g′]_{p}). θ is well-defined by Lemma 2.8 (1).

3.2 Lemma.

(i) θ : F_{2} → E is a monoid-epimorphism. (ii) The inverse image of H under θ : F_{2} → E is G_{2}. (iii) The restriction of θ to G_{2} is a group epimorphism θ:G_{2} →H with |ker(θ)|=p^{p}. (iv) ψ : E → F_{1} is a monoid epimorphism and ψ restricted to H is a group-

epimorphism ψ : H → G_{1}.

Proof. (i) follows from Lemma 2.8 (3) and (ii) from Fact 2.7. (iii) follows from Lemma 2.8 (4). Finally, (iv) holds because every function on ℤ_{p} is a polynomial function and every permutation of ℤ_{p} is a polynomial permutation. □

4. Sylow subgroups of H

We will first determine the Sylow p-groups of H. The Sylow p-groups of G_{n} for n ≥ 2 are obtained in the next section as the inverse images of the Sylow p-groups of H under the epimorphism G_{n} → H.

4.1 Lemma. Let C_{0} be the subgroup of S_{p} generated by the p-cycle (0 1 2 … p−1). Then one Sylow p-subgroup of H is

S = {(f, f′) ∈ H | f ∈ C_{0}, f′ = 1},

where f′ = 1 means the constant function 1. The normalizer of S in H is

N_{H}(S) = {(g, g′) | g ∈ N_{S_{p}}(C_{0}), g′ a non-zero constant }.

Proof. As |H| = p!(p − 1)^{p}, and S is a subgroup of H of order p, S is a Sylow p-group of H. Conjugation of (f, f′) ∈ S by (g, g′) ∈ H (using the fact that f′ = 1) gives

(g, g′)^{−1}(f, f′)(g, g′) = (g^{−1}, \frac{1}{g′ ∘ g^{−1}} )(f ∘ g, g′) = (g^{−1} ∘ f ∘ g, \frac{g′}{g′ ∘ g^{−1} ∘ f ∘ g})

The first coordinate of (g, g′)^{−1}(f, f′)(g, g′) being in C_{0} for all (f, f′) ∈ S is equivalent to g ∈ N_{S_{p}}(C_{0}). The second coordinate of (g, g′)^{−1}(f, f′)(g, g′) being the constant function 1 for all (f, f′) ∈ S is equivalent to

∀x ∈ ℤ_{p} g′(x) = g′(g^{−1}(f(g(x))),

which is equivalent to g′ being constant on every cycle of g^{−1}fg, which is equivalent to g′ being constant on ℤ_{p}, since f can be chosen to be a p-cycle. □

4.2 Lemma. Another way of describing the normalizer of S in H is

N_{H}(S) = {(g, g′) ∈ H | ∃k ≠ 0 ∀a, b g(a) − g(b) = k(a − b); g′ a non-zero constant}.

Therefore, |N_{H}(S)| = p(p − 1)^{2} and [H : N_{H}(S)] = (p − 1)!(p − 1)^{p−2}.

Proof. Let σ = (0 1 2 … p − 1) and g ∈ S_{p} then

gσg^{−1} = (g(0) g(1) g(2) … g(p − 1))

Now g ∈ N_{S_{p}}(C_{0}) if and only if, for some 1 ≤ k < p gσg^{−1} = σ^{k}, i.e.,

(g(0) g(1) g(2) … g(p − 1)) = (0 k 2k … (p − 1)k),

all numbers taken mod p. This is equivalent to g(x + 1) = g(x) + k or

g(x + 1) − g(x) = k

and further equivalent to g(a) − g(b) = k(a − b). Thus k and g(0) determine g ∈ N_{S_{p}}(C_{0}), and there are (p − 1) choices for k and p choices for g(0). Together with the (p − 1) choices for the non-zero constant g′ this makes p(p − 1)^{2} elements of N_{H}(S). □

4.3 Corollary. There are (p − 1)!(p − 1)^{p−2} Sylow p-subgroups of H.

4.4 Theorem. The Sylow p-subgroups of H are in bijective correspondence with pairs (C, φ̄), where C is a cyclic subgroup of order p of S_{p}, φ : ℤ_{p} → ℤ_{p} \ {0} is a function and φ̄ is the class of φ with respect to the equivalence relation of multiplication by a non-zero constant. The subgroup corresponding to (C, φ̄) is

S_{(C,φ̄)} = \{(f, f′) ∈ H | f ∈ C, f′(x) = \frac{φ(f(x))}{φ(x)}\}

Proof. Observe that each S_{(C,φ̄)} is a subgroup of order p of H. Different pairs (C, φ̄) give rise to different groups: Suppose S_{(C,φ̄)} = S_{(D,ψ̄)}. Then C = D and for all x ∈ ℤ_{p} and for all f ∈ C we get

\frac{φ(f(x))}{φ(x)} = \frac{ψ(f(x))}{ψ(x)}.

As C is transitive on ℤ_{p} the latter condition is equivalent to

∀x, y ∈ ℤ_{p} \frac{ψ(x)}{φ(x)} = \frac{ψ(y)}{φ(y)},

which means that φ = kψ for a nonzero k ∈ ℤ_{p}.

There are (p−2)! cyclic subgroups of order p of S_{p}, and (p−1)^{p−1} equivalence classes φ̄ of functions φ : ℤ_{p} → ℤ_{p} \ {0}. So the number of pairs (C, φ̄) equals (p − 1)!(p − 1)^{p−2}, which is the number of Sylow p-groups of H, by the preceding corollary. □

4.5 Proposition. If p is an odd prime then the intersection of all Sylow p-subgroups of H is trivial, i.e.,

⋂_{(C,\overline{φ})} S_{(C,\overline{φ})} = \{(ι, 1)\}.

If p = 2 then |H| = 2 and the intersection of all Sylow 2-subgroups of H is H itself.

Proof. Let p be an odd prime, and let (f, f′) ∈ ⋂_{(C,\overline{φ})} S_{(C,\overline{φ})}. Suppose f is not the identity function and let k ∈ ℤ_{p} such that f(k) ≠ k.

Note that φ in (C, \overline{φ}) is arbitrary, apart from the fact that 0 is not in the image. Therefore, and because p ≥ 3, among the various φ there occur functions ϑ and η with ϑ(k) = η(k) and ϑ(f(k)) ≠ η(f(k)). Now (f, f′) ∈ S_{(D,\overline{ϑ})} ∩ S_{(E,\overline{η})} for any cyclic subgroups D and E of S_{p} of order p.

Therefore

\frac{ϑ(f(k))}{ϑ(k)} = f′(k) = \frac{η(f(k))}{η(k)},

and hence ϑ(f(k)) = η(f(k)), a contradiction. Thus f is the identity and therefore f′ = 1.

If p = 2 then |H| = 2 and therefore the one and only Sylow 2-subgroup of H is H. □

In the case p ≥ 5, the lemma above can be proved in a simpler way: There is more than one cyclic group of order p, so for (f, f′) ∈ ⋂_{(C,\overline{φ})} S_{(C,\overline{φ})}, there are distinct cyclic groups D and E of order p with f ∈ D ∩ E. Therefore f has to be the identity.

5. Sylow subgroups of G_{n} and of the projective limit G

Again we consider the projective system of finite groups

G_{1} ⟵^{ψ} H ⟵^{θ} G_{2} ⟵^{π_{2}} … ⟵^{π_{n−1}} G_{n} ⟵^{π_{n}}

where (G_{n}, ∘) is the group of polynomial permutations on ℤ_{p^{n}} (with respect to composition of functions) and H is the group defined in section 3. Let G = \underleftarrow{lim} G_{n} be the projective limit of this system. Recall that a Sylow p-group of a pro-finite group is defined as a maximal group consisting of elements whose order in each of the finite groups in the projective system is a power of p.

5.1 Theorem.

(i) Let (G_{n}, ∘) be the group of polynomial permutations on ℤ_{p^{n}} with respect to composition. If n ≥ 2 there are (p − 1)!(p − 1)^{p−2} Sylow p-groups of G_{n}. They are the inverse images of the Sylow p-groups of H (described in Theorem 4.4) under the canonical projection π : G_{n} → H, with π = θπ_{2} … π_{n−1}.

(ii) Let G = \underleftarrow{lim} G_{n}. There are (p − 1)!(p − 1)^{p−2} Sylow p-groups of G, which are the inverse images of the Sylow p-groups of H (described in Theorem 4.4) under the canonical projection π : G → H.

Proof. In the projective system G_{1} ⟵^{ψ} H ⟵^{θ} G_{2} ⟵^{π_{2}} … ⟵^{π_{n−1}} G_{n} the kernel of the group epimorphism G_{n} → H is a finite p-group for every n ≥ 2, because for n ≥ 2 the kernel of π_{n} : G_{n+1} → G_{n} is of order p^{β(n+1)} by Corollary 2.10, and the kernel of θ : G_{2} → H is of order p^{p} by Lemma 3.2 (iii). So the Sylow p-groups of G_{n} for n ≥ 2 are just the inverse images of the Sylow p-groups of H and, likewise, the Sylow p-groups of the projective limit G are just the inverse images of the Sylow p-groups of H, whose number was determined in Corollary 4.3. □

If we combine this information with the description of the Sylow p-groups of H in Theorem 4.4 we get the following explicit description of the Sylow p-groups of G_{n}. Recall that [f]_{p^{n}} denotes the function induced on ℤ_{p^{n}} by the polynomial f in ℤ[x] (or in ℤ_{p^{m}}[x] for some m ≥ n).

5.2 Corollary. Let n ≥ 2. Let G_{n} be the group (with respect to composition) of polynomial permutations on ℤ_{p^{n}}. The Sylow p-groups of G_{n} are in bijective correspondence with pairs (C, φ̄), where C is a cyclic subgroup of order p of S_{p}, φ : ℤ_{p} → ℤ_{p} \ {0} is a function and φ̄ its class with respect to the equivalence relation of multiplication by a non-zero constant. The subgroup corresponding to (C, φ̄) is

S_{(C,φ̄)} = \{[f]_{p^{n}} ∈ G_{n} | [f]_{p} ∈ C, [f′]_{p}(x) = \frac{φ([f]_{p}(x))}{φ(x)}\}.

Example. A particularly easy to describe Sylow p-group of G_{n} is the one corresponding to (C, φ) where φ is a constant function and C the subgroup of S_{p} generated by (0 1 2 … p−1). It is the inverse image of S defined in Lemma 4.1 and it consists of the functions on ℤ_{p^{n}} induced by polynomials f such that the formal derivative f′ induces the constant function 1 on ℤ_{p} and the function induced by f itself on ℤ_{p} is a power of (0 1 2 … p − 1).

Combining Theorem 5.1 with Proposition 4.5 we obtain the following description of the intersection of all Sylow p-groups of G_{n} for odd p.

5.3 Corollary. Let p be an odd prime.

(i) For n ≥ 2 the intersection of all Sylow p-groups of G_{n} is the kernel of the projection π : G → H.

(ii) Likewise, the intersection of all Sylow p-groups of G is the kernel of the

canonical epimorphism of G onto H. (iii) The intersection of all Sylow p-groups of G_{n} (n ≥ 2) can also be described as

the normal subgroup

N = {[f]_{p^{n}} ∈ G_{n} | [f]_{p} = ι, [f′]_{p} = 1},

where ι denotes the identity function on ℤ_{p}. Its order is p^{p}p^{∑_{k=3}^{n} β(k)} and its index in G_{n} (for n ≥ 2) is

[G_{n} : N] = p!(p − 1)^{p}

(iv) Likewise, the index of the intersection of all Sylow p-subgroups of G in G is p!(p − 1)^{p}.

Proof. (i) and (ii) follow immediately from Theorem 5.1 and Proposition 4.5. To see (iii), let π be the projection from G_{n} to H (that is π = θπ_{2} … π_{n−1}). Then N is the inverse image of {(ι, 1)}, the identity element of H, under π, and is therefore the intersection of the Sylow p-groups of G_{n} by (i). As the kernel of a group homomorphism, N is a normal subgroup.

The order of N is the order of the kernel of π, which is the product of p^{p} (the order of the kernel of θ) and p^{β(k)} (the order of the kernel of π_{k−1}) for 3 ≤ k ≤ n. Finally, the index of the kernel of the homomorphism of G_{n} or G onto H is the order of H which is p!(p − 1)^{p}. □

Acknowledgement. The authors wish to thank W. Herfort for stimulating discussuions.

References

[1] M. Bhargava, P -orderings and polynomial functions on arbitrary subsets of Dedekind rings, J. Reine Angew. Math., 490 (1997), pp. 101–127.

[2] J. V. Brawley and G. L. Mullen, Functions and polynomials over Galois rings, J. Number Theory, 41 (1992), pp. 156–166.

[3] L. Carlitz, Functions and polynomials (mod p^{n}), Acta Arith., 9 (1964), pp. 67–78.

[4] Z. Chen, On polynomial functions from ℤ_{n_{1}} ×ℤ_{n_{2}} ×⋯×ℤ_{n_{r}} to ℤ_{m}, Discrete Math., 162 (1996), pp. 67–76.

[5] S. Frisch, When are weak permutation polynomials strong?, Finite Fields Appl., 1 (1995), pp. 437–439.

[6] , Polynomial functions on finite commutative rings, in Advances in commutative ring theory (D.E. Dobbs et al. eds), vol. 205 of Lecture Notes in Pure and Appl. Math., Dekker, New York, 1999, pp. 323–336.

[7] J. Jiang, A note on polynomial functions over finite commutative rings, Adv. Math. (China), 39 (2010), pp. 555–560.

[8] J. J. Jiang, G. H. Peng, Q. Sun, and Q. Zhang, On polynomial functions over finite commutative rings, Acta Math. Sin. (Engl. Ser.), 22 (2006), pp. 1047–1050.

[9] G. Keller and F. R. Olson, Counting polynomial functions (mod p^{n}), Duke Math. J., 35 (1968), pp. 835–838.

[10] A. J. Kempner, Polynomials and their residue systems, Trans. Amer. Math. Soc., 22 (1921), pp. 240–266, 267–288.

[11] N. P. Liu and J. J. Jiang, Polynomial functions in n variables over a finite commutative ring, Sichuan Daxue Xuebao, 46 (2009), pp. 44–46.

[12] B. R. McDonald, Finite Rings with Identity, Dekker, 1974.

[13] A. A. Nechaev, Polynomial transformations of finite commutative local rings of principal ideals, Math. Notes, 27 (1980), pp. 425–432. transl. from Mat. Zametki 27 (1980) 885-897, 989.

[14] W. Nöbauer, Gruppen von Restpolynomidealrestklassen nach Primzahlpotenzen, Monatsh. Math., 59 (1955), pp. 194–202.

[15] , Polynomfunktionen auf primen Restklassen, Arch. Math. (Basel), 39 (1982), pp. 431–435.

[16] I. G. Rosenberg, Polynomial functions over finite rings, Glas. Mat., 10 (1975), pp. 25–33.

[17] Q. Wei and Q. Zhang, On strong orthogonal systems and weak permutation polynomials over finite commutative rings, Finite Fields Appl., 13 (2007), pp. 113–120.

[18] Q. J. Wei and Q. F. Zhang, On permutation polynomials in two variables over ℤ/p^{2}ℤ, Acta Math. Sin. (Engl. Ser.), 25 (2009), pp. 1191–1200.

[19] Q. Zhang, Polynomial functions and permutation polynomials over some finite commutative rings, J. Number Theory, 105 (2004), pp. 192–202.

S. F. Institut für Mathematik A Technische Universität Graz Steyrergasse 30 A-8010 Graz, Austria

frisch@tugraz.at

D. K.

Institut für Mathematik B Technische Universität Graz Steyrergasse 30 A-8010 Graz, Austria

                             krenn@math.tugraz.at