COHOMOLOGICAL INVARIANTS

OF MAPPING CLASS GROUPS

Stewart Priddy and Yining Xia

4/3/1995

Introduction

The mapping class group Γ_{g} is defined to be the group of path components of orientation preserving diffeomorphisms of an oriented closed surface S_{g} of genus g. In this paper, we study for a given prime p a series of cohomological integral invariants p_{r}(Γ_{g}) of mapping class groups.

Let Γ be a group of finite virtual cohomological dimension. The first invariant p_{1}(Γ) of this series is the well-known Yagita invariant p(Γ) (see [5] or [9]), which is defined via restriction maps in cohomology to its cyclic subgroups of order p. The invariant p_{r}(Γ) (r > 1) is a natural generalization defined in the same way via restriction maps in cohomology to its elementary abelian p-subgroups of rank r (see Section 1).

The cohomological invariant p_{r}(Γ) shares a basic property of the p-period which states that p_{r}(H) divides p_{r}(Γ) if H is a subgroup of Γ. Thus, the invariant could be considered as a type of local obstruction for a group being a subgroup of Γ. It also provides a lower bound for the dimension of certain Γ-spaces of homotopy type (S^{m})^{k}. As our main result (Theorem 3.4), we show the calculation of p_{r}(Γ_{g}) can be made using some geometric and representational methods for Γ_{g}. It appears that the invariant p_{r}(Γ_{g}) becomes a useful tool in practice for studying subgroup structures of Γ_{g}.

The first invariant p_{1}(Γ_{g}) (namely the Yagita invariant) has been studied extensively by Thomas [7], Glover, Mislin and the second author in [4], [5] and [8]. We will mainly consider p_{r}(Γ_{g}) for r > 1 in this paper. After some preliminary results are given in Section 1, we establish a lower bound for p_{r}(Γ_{g}) in Section 2. Section 3 is devoted to establishing an upper bound.

We would like to thank the referee for his or her valuable suggestions.

Section 1: Definition and basic properties of the invariant p_{r}(Γ)

Let Γ be a group of finite virtual cohomological dimension and π ⊂ Γ any elementary abelian p-subgroup of rank r, namely, π isomorphic to (Z/p)^{r} for a prime p and a positive integer r. It is easy to see that the image Im(H^{k}(Γ; ℤ) ⟶ H^{k}(π; ℤ)) of the restriction map in cohomology is non-zero for some degree k > 0. For any cohomology element x ∈ Im(H^{k}(Γ; ℤ) ⟶ H^{k}(π; ℤ)), let m_{x} denote the degree of x. Define m(π) the greatest common divisor of m_{x} where x ranges over all nontrivial cohomology elements of Im(H^{k}(Γ; ℤ) ⟶ H^{k}(π; ℤ)). The invariant p_{r}(Γ) with respect to prime p and positive integer r is defined to be the least common multiple of values m(π), where π ranges over all elementary abelian p-subgroups of rank r. Since Γ has a torsion-free normal subgroup of finite index and p_{r}(G) for G finite is finite, thus p_{r}(Γ) is finite. We use the convention that p_{r}(Γ) = 1 if Γ does not contain any (ℤ/p)^{r}. It is easy to see that the first invariant p_{1}(Γ) is the same as the Yagita invariant p(Γ) defined in [5]. Also, the following three propositions are clear from the definition above; we state them without proof.

Proposition 1.1. p_{r}(E) = 2 if E is an abelian group containing an elementary abelian p-subgroup of rank r.

Proposition 1.2. p_{r+1}(Γ) divides p_{r}(Γ); In particular, p_{r}(Γ) divides the Yagita invariant p_{1}(Γ).

Proposition 1.3. If H is a subgroup of Γ, then p_{r}(H) | p_{r}(Γ).

Next proposition provides an upper bound for the invariant p_{r}(Γ).

Proposition 1.4. Let p be a prime and Γ a group of finite virtual cohomological dimension which contains (ℤ/p)^{r}. Then the invariant p_{r}(Γ) divides 2(p − 1)p^{k} for some k.

Proof. It follows from Proposition 1.2 and Lemma 1.1 of [5].

As with the Yagita invariant p(Γ), the invariant p_{r}(Γ) provides a lower bound for the dimension of a complex, which admits a certain type of action of Γ. This is related to mixed type of spherical space form problems studied by Stark [6]. One has

Proposition 1.5. Let X be a finite dimensional, admissible, proper Γ complex of homotopy type (S^{n})^{k} (see [1] for terminology). If Γ acts on H^{∗}(X) trivially in a way that the stabilizer of any point x ∈ X does not contain (ℤ/p)^{r}, then m + 1 is a multiple of the invariant p_{r}(Γ).

Proof. Consider two spectral sequences which converge to the equivalent cohomology H_{Γ}^{∗}(X)

E_{2}^{a,b} = H^{a}(Γ; H^{b}(X)) ⟹ H_{Γ}^{a+b}(X)  (1)

and

E_{1}^{a,b} = ⊕_{δ∈∑_{a}}H^{b}(Γ_{δ}) ⟹ H_{Γ}^{a+b}(X)  (2)

where ∑_{a} is a set of representatives for the a-cells of X mod Γ and Γ_{δ} is the stabilizer of δ. It suffices to show for any elementary abelian p-subgroup π ⊂ Γ of rank r, there exists a non-trivial differential d_{m+1} at E_{m+1}^{0,m} in the spectral sequence which restricts from (1) to

E_{2}^{a,b} = H^{a}(π; H^{b}(X)) ⟹ H_{π}^{a+b}(X)

If not, the Krull dimension of H_{π}^{∗}(X) is equal to r, which contradicts the calculation from the spectral sequence which restricts (2) to

E_{1}^{a,b} = ⊕_{δ_{a}}H^{b}(Γ_{δ}) ⟹ H_{π}^{a+b}(X)

since no stabilizer stab_{Γ}(x) contains (ℤ/p)^{r}.

In section 3, we will use a variation of the last proposition for estimating p_{r}(Γ_{g}) by means of a geometric representation of Γ_{g}.

Section 2: A lower bound of p_{r}(Γ_{g})

A basic invariant of an effective orientation-preserving finite group G action on S_{g} is the set δ(G) of its fixed point data [3]. In the general case of an oriented surface with orientation-preserving action, the isotropy groups are all cyclic, and the set δ(G) of fixed point data is given as a set (with multiplicities) of conjugacy classes in G of generators for these isotropy groups G_{x}, one for each singular orbit. To this end let {x_{i}} be a set of representatives of the singular orbits of G. If G is abelian, δ(G) becomes an unordered set of nontrivial, not necessarily distinct elements of G with product the identity. We describe this data by

δ(G) = ⟨g, G | g_{1}, …, g_{q}⟩

where g is the genus of the surface S_{g}, q the number of singular orbits of the G action and g_{i} a generator of stab_{G}(x_{i}) which acts on the tangent space at x_{i} by rotation through 2π/|stab_{G}(x_{i})|. The definition of fixed point data of a G action on S_{g} does not depend on the isotopy class of G. Thus, for any finite abelian subgroup G ⊂ Γ_{g}, we may define the fixed point data of G as

δ(G) = ⟨g, G | g_{1}, …, g_{q}⟩ .

A modification of the proof of Lemma 2.2 in [5] gives the following Lemma which is used later. We leave details to the reader.

Lemma 2.1. Let p be a prime and G an abelian p-group. Then the collection

⟨g, G | g_{1}, …, g_{q}⟩

can be realized as the fixed point data of G in Γ_{g} if and only if the following three conditions are satisfied.

(i) ∏_{i=1}^{q} g_{i} = 1 (ii) The Riemann-Hurwitz formula 2g −2 = |G|(2h−2)+|G| ∑(1−1/p^{t_{i}}) holds

for some h ≥ 0, where p^{t_{i}} = |⟨g_{i}⟩| (iii) The rank of the abelian p-subgroup H = G/⟨g_{1}, …, g_{q}⟩ ≤ 2h.

Definition. Let p be a prime. We say that an integer g > 1 satisfies the (p_{r})- condition if and only if g is of the form lp^{α} + 1 with l prime to p, α ≥ r, and 2l = p(2h−2)+k(p−1) for some integers h > 0, k ≥ 0 with k ≠ 1 and 2h+k−3 ≥ r.

Remark 1. The (p_{1})-condition above is the same as the p-condition in [5]. For a fixed prime p and α > 0 there are obviously only finitely many genera g of the form lp^{α} + 1 with l prime to p which do not satisfy the (p_{r})-condition. Also, the (p_{r})-condition has a geometric interpretation as follows.

Let q be denoted as the natural quotient map from Diff_{+}(S_{g}) to Γ_{g}, where Diff_{+}(−) is the group of all orientation preserving diffeomorphisms.

Lemma 2.2. Let p be an odd prime and assume that g = lp^{α} + 1 with l prime to p and α > 0. Then g satisfies the (p_{r})-condition implies that

(i) Γ_{g} contains an elementary abelian p subgroup π of rank r and there is a lift of the π via the map q acting on S_{g} freely.

(ii) Γ_{g} contains an elementary abelian p-subgroup π of rank r so that for every order p element x ∈ π there exists an abelian subgroup K_{x} ⊂ Γ_{g} of order p^{α+1} and an element k_{x} ∈ K_{x} satisfying k_{x}^{p^{α−r+1}} = x, and a lift of the cyclic group C_{p^{α−r+2}} = ⟨k_{x}⟩ via the map q acting on S_{g} has singularities at most of order p.

Proof. It is well-known from the basic covering space theory that a surjection f : π_{1}(B) ⟶ G gives rise a covering map p : E ⟶ B, where B and E are path connected and locally path connected. The group of deck transformations of E is isomorphic to G. Let h = lp^{α−r} + 1. Then the p_{r}-condition implies 2g − 2 = p^{r}(2lp^{α−r}) = p^{r}(2h − 2) and h ≥ l + 1 ≥ p(h − 1) + k(p − 1)/2 + 1 ≥ 2h + k − 1 ≥ r. Thus the Euler characteristic of S_{g} is p^{r} times that of S_{h} and we can construct a π = (ℤ/p)^{r} free action on S_{g} via a surjection π_{1}(S_{h}) ⟶ π by mapping the first r ā_{i}’s to generators of π and the remaining ā_{i}’s and all b̄_{i}’s to 1, where ā_{1}, …, ā_{h}; b̄_{1}, …, b̄_{h} are generators of π_{1}(S_{h}), and h = lp^{α−r} + 1. We show (ii) next. Since g satisfies the p_{r}-condition, one has 2g − 2 = p^{r}(2h′ − 2) + k′p^{r}(1 − 1/p) for some integers h′ = p^{α−r+1}(h − 1) + 1 and k′ = p^{α−r+1}k. Let π = (ℤ/p)^{r} = ⟨a_{1}, …, a_{r}⟩ be an abstract elementary abelian p-group. The collection

⟨g, π | a_{1}, …, a_{1}, …, a_{k−1}, …, a_{k−1}, (a_{1} … a_{k−1})^{−1} … (a_{1} … a_{k−1})^{−1}⟩

(with p^{α−r+1} each of a_{1}, a_{2}, ..., a_{k−1}) can be realized as the fixed point data of π according to Lemma 2.1 because of 2h + k − 3 ≥ r. For any non-trivial element x ∈ π, let K_{x} denote an abelian group of order p^{α+1} so that K_{x} contains π as a subgroup and there exists an element k_{x} of K_{x} satisfying k_{x}^{p^{α−r+1}} = x. We claim that K_{x} is isomorphic to a subgroup of the mapping class group Γ_{g}. This is because one can realize

⟨g, K_{x} | a_{1}, …, a_{k−1}, (a_{1} … a_{k−1})^{−1}⟩

as the fixed point data of a K_{x} action by Lemma 2.1. For example, a K_{x} action on S_{g} can be constructed via a surjection f : π_{1}(S_{h} − {k}) ⟶ K_{x} such that f(x_{i}) = a_{i} (1 ≤ i ≤ k − 1), f(x_{k}) = (a_{1} … a_{k−1})^{−1}, f(ā_{1}) = k_{x}, f(b̄_{1}) = 1, where x_{1}, …, x_{k}; ā_{1}, …, ā_{h}; b̄_{1}, …, b̄_{h} are generators of π_{1}(S_{h} − {k}). The (p_{r})-condition guarantees such map exists.

We need the following description of the normalizer N(−) and centralizer C(−) of an abelian p-subgroup G ⊂ Γ_{g} in order to show Lemma 2.3. It is well-known that the quotients N(G)/G and C(G)/G are isomorphic to subgroups of the mapping class group Γ_{h}^{k} with k punctures, where S_{h} = S_{g}/G with k branch points. Consider a natural map λ : Γ_{h}^{k} ⟶ Aut(H_{1}(S_{h} − {k}; ℤ)) given by the induced map on H_{1}(S_{h} − {k}; ℤ). Then λ(N(G)/G) consists of all w ∈ Aut(H_{1}(S_{h} − {k}; ℤ)) which preserve the symplectic form such that fw = βf, where f : H_{1}(S_{h} − {k}; ℤ) ⟶ G, called the classifying map, is determined by the given G action , β ∈ Aut(G). Further, λ(C(G)/G) consists of all w ∈ Aut(H_{1}(S_{h} − {k}; ℤ)) which preserve the symplectic form such that fw = f. The proof of statement above follows from the basic covering space theory. We omit it here.

Lemma 2.3.

(i) Let π = ⟨y_{1}, … y_{r}⟩ ⊂ Γ_{g} be the elementary abelian p subgroup of rank r constructed as in Lemma 2.2 (i). For any y_{i} (1 ≤ i ≤ r), there is an element g_{i} ∈ Γ_{g} such that g_{i}y_{i}g_{i}^{−1} = y_{i}^{d} and g_{i}y_{j}g_{i}^{−1} = y_{j} (i ≠ j), where d is a generator of (ℤ/p)^{∗}.

(ii) Let K_{x} = ⟨k_{x}, y_{2}, …, y_{r}⟩ ⊂ Γ_{g} be the abelian subgroup constructed as in Lemma 2.2 (ii). There is an element g_{x} ∈ Γ_{g} such that g_{x}k_{x}g_{x}^{−1} = k_{x}^{p+1} and g_{x}y_{i}g_{x}^{−1} = y_{i} (2 ≤ i ≤ r).

Proof. By viewing the normalizer structure N(π)/π described above, one needs to choose f as given in the example of the proof of Lemma 2.2 (i) and β ∈ Aut(π) which maps y_{i} to y_{i}^{d} and y_{j} to y_{j} for i ≠ j. The existence of such g_{i} follows. For the proof of (ii), again, by viewing the normalizer structure N(K_{x})/K_{x}, one needs to choose f as given in the example of the proof of Lemma 2.2 (ii) and β ∈ Aut(K_{x}) which maps k_{x} to k_{x}^{p+1} and y_{i} to y_{i}(2 ≤ i ≤ r). The existence of g_{x} is obvious now.

Lemma 2.4. p_{r}(N(π)) is a multiple of 2(p − 1).

Proof. Let u ∈ H^{∗}(N(π); ℤ) be a cohomology element whose restriction to H^{∗}(π; ℤ) is non-trivial. Write Res|_{π}(u) = ∑ a_{i_{1},...,i_{r}}y_{1}^{i_{1}} … y_{r}^{i_{r}} ∈ 𝔽_{p}[y_{1}, …, y_{r}], where a_{i_{1},...,i_{r}} ≠ 0. Assume that one of powers i_{j} (1 ≤ j ≤ r), say i_{1}, is not a multiple of p−1. Note that Res|_{π}(u) = Res|_{π}(g_{1}ug_{1}^{−1}), i.e., ∑ a_{i_{1},...,i_{r}}y_{1}^{i_{1}} … y_{r}^{i_{r}} = ∑ a_{i_{1},...,i_{r}}d^{i_{1}}y_{1}^{i_{1}} … y_{r}^{i_{r}}. d^{i_{1}} ≠ 1 mod(p) implies a_{i_{1},...,i_{r}} = 0 mod(p), a contradiction. This shows that all i_{j}’s are multiples of p − 1, i.e., the degree of u is a multiple of 2(p − 1).

Lemma 2.5. p_{r}(N(K_{x})) is a multiple of 2p^{α−r+1}.

Proof. Let π ⊂ K_{x} ⊂ N(K_{x}) be the elementary abelian p subgroup and u ∈ H^{∗}(N(K_{x}); ℤ) be a cohomology element whose restriction to H^{∗}(π; ℤ) is non-trivial. As before, write Res|_{π}(u) = ∑ a_{i_{1},...,i_{r}}y_{1}^{i_{1}} … y_{r}^{i_{r}} ∈ 𝔽_{p}[y_{1}, …, y_{r}], where a_{i_{1},...,i_{r}} ≠ 0. Assume that one of i_{j}’s, say i_{1}, is not a multiple of p^{α−r+1}. One can write Res|_{K_{x}}(u) = ∑ ā_{i_{1},...,i_{r}}y_{1}^{i_{1}} … y_{r}^{i_{r}} ∈ 𝔽_{p^{α−r+2}}[y_{1}, …, y_{r}]. Therefore, Res|_{K_{x}}(u) = Res|_{K_{x}}(g_{x}ug_{x}^{−1}) implies that ∑ ā_{i_{1},...,i_{r}}y_{1}^{i_{1}} … y_{r}^{i_{r}} = ∑ ā_{i_{1},...,i_{r}}(p + 1)^{i_{1}}y_{1}^{i_{1}} … y_{r}^{i_{r}} in 𝔽_{p^{α−r+2}}[y_{1}, …, y_{r}]. This says ā_{i_{1},...,i_{r}} is a multiple of p, i.e., a_{i_{1},...,i_{r}} = 0 mod(p), a contradiction. This amounts to saying all i_{j}’s (1 ≤ j ≤ r) are multiples of p^{α−r+1}, which gives the result.

Combining Lemmas 2.4 and 2.5, we conclude

Proposition 2.6. Let p be an odd prime and assume that g = lp^{α} + 1 with l prime to p and α > 0. If g satisfies the (p_{r})-condition, then the invariant p_{r}(Γ_{g}) is a multiple of 2(p − 1)p^{α−r+1}.

Section 3: An upper bound of p_{r}(Γ_{g})

Let π ⊂ Γ_{g} be an elementary abelian p-subgroup of rank r. Let θ : Γ_{g} ⟶ GL_{2g}(ℂ) denote the representation given by the action of Γ_{g} on H_{1}(S_{g}; ℂ). Note that θ factors through GL_{2g}(ℚ). Let {⟨z_{i}⟩ | i = 1, …, m} denote the m = p^{r−1} + p^{r−2} +⋯+1 subgroups of π of order p. Then by rational representation theory (see [2], Section 39), θ |π = ∑_{i=0}^{m} n_{i}δ_{i} with δ_{0} the trivial one-dimensional representation and δ_{i}(i > 0) the (p − 1)-dimensional irreducible reduced regular representation corresponding to ⟨z_{i}⟩. Here ⟨z_{i}⟩ ranges over the m = p^{r−1} +p^{r−2} +⋯+1 subgroups of π of order p.

We study Chern classes c_{i}(θ) ∈ H^{2i}(Γ_{g}; ℤ) restricting to H^{2i}(π; ℤ). By calculating the trace and using the Lefschetz fixed point theorem, one obtains an equation system consisting of m + 1 equations. Namely,

A ( n_{0} ; n_{1} ; ⋮ ; n_{m} ) = ( 2g ; 2 − t_{1} ; ⋮ ; 2 − t_{m} )

where t_{i} is the number of the fixed points of ⟨z_{i}⟩ ⊂ π of order p on the orbit S_{g}/⟨z_{i}⟩. It is interesting that the coefficient matrix A_{m+1,m+1} has the following simple property.

Lemma 3.1. A^{2} = p^{r}I_{m+1}.

Proof. Write

A = ( \begin{matrix}1 & B \\ C & D \end{matrix} )

where B = (p − 1, …, p − 1),

C = ( 1 ⋮ 1 )

and D is a (m × m) symmetric matrix with entry either p − 1 or -1. This is because of trδ_{i}(z_{j}) = p − 1 or −1 depending on i, j and trδ_{i}(z_{j}) = trδ_{j}(z_{i}). Actually, it is easy to see that in each row of D (and each column) there are p^{r−1} entries -1 and p^{r−2} + p^{r−3} + ⋯ + 1 entries p − 1. Therefore, at each diagonal entry of A^{2}, one has a_{1,1} = 1 + m(p − 1) = p^{r} and a_{i,i} = p^{r−1} + (p^{r−2} + … 1)(p − 1)^{2} + (p − 1) = p^{r} for i > 1. If multiplying two different rows of D, one observes that there are 2p^{r−2} many (p−1)×(−1), p^{r−1}−p^{r−2} many (−1)×(−1) and p^{r−3}+⋯+1 many (p−1)×(p−1). Thus, at non-diagonal entry of A^{2} one has a_{1,j} = p − 1 + (p − 1)[(p − 1)(p^{r−2} + p^{r−3} + ⋯ + 1) − p^{r−1}] = 0 for j > 1, a_{i,1} = 1 + (p − 1)(p^{r−2} + ⋯ + 1) − p^{r−1} = 0 for i > 1 and a_{i,j} = p − 1 − 2p^{r−2}(p − 1) + p^{r−1} − p^{r−2} + (p^{r−3} + ⋯ + 1)(p − 1)^{2} = 0 for i, j > 0. □

It is well known that the total Chern class C = Σc_{i} satisfies

C(θ |π) = Π(1 − y_{i}^{p−1})^{n_{i}}

where y_{i} ∈ H^{2}(π; ℤ) corresponds to z_{i}. Hence we determine gcd{n_{i}}. In the expression

( n_{0} ; n_{1} ; ⋮ ; n_{m} ) = A^{−1} ( 2g ; 2 − t_{1} ; ⋮ ; 2 − t_{m} )

i.e.,

( n_{0} ; n_{1} ; ⋮ ; n_{m} ) = (1/p^{r})A ( 2g ; 2 − t_{1} ; ⋮ ; 2 − t_{m} )

one has for i > 0 that

gcd{n_{i}} = (1/p^{r})gcd{2g + ∑_{j}(p − 1)(2 − t_{j}) − ∑_{k}(2 − t_{k})}

where j (resp. k) ranges over complementary sets of m − p^{r−1} (resp. p^{r−1}) integers whose union is {1, ..., m}. Thus

gcd{n_{i}} = (1/p^{r})gcd{2g − 2 + p^{r−1}t − p ∑ t_{i}}

where t is the number of singular points of S_{g}/π and ∑ t_{i} = p^{r−1}t.

The following Lemma is a generalization of Proposition 4.3 in [5]. It is easy to see that its proof can be carried out in the same way.

Lemma 3.2. Let g > 1 and p a prime. Let π be an elementary abelian p-subgroup of rank r, with associated Riemann-Hurwitz formula 2g −2 = p^{r}(2h−2)+tp^{r−1}(p−1). Then there exists a cohomology element e ∈ H^{6(g−h)−2t}(Γ_{g}; ℤ) whose restriction to H^{6(g−h)−2t}(π; ℤ) is non-trivial.

Next, we combine Lemmas 3.1 and 3.2 to show the main theorem of this section.

Theorem 3.3. Suppose g = lp^{α} + 1 with l prime to p and p an odd prime. Then for any r ≤ α + 1, the invariant p_{r}(Γ_{g}) divides 2(p − 1)p^{α−r+1}.

Proof. It suffices to show that for any π ⊂ Γ_{g} there is a cohomology element

e ∈ H^{2j(π)p^{α−r+1}}(Γ_{g}; ℤ)

whose restriction to H^{2j(π)p^{α−r+1}}(π; ℤ) is non-trivial where j(π) is prime to p.

We prove that either

gcd{n_{i}} ≢ 0 mod(p^{α−r+2})

or

6(g − h) − 2t ≢ 0 mod(p^{α−r+2})

holds.

If gcd{n_{i}} ≡ 0 mod(p^{α−r+2}), then for any 1 ≤ q ≤ m

n_{q} = (1/p^{r})(2lp^{α} + p^{r−1}t − p ∑ t_{i}) ≡ 0 mod(p^{α−r+2})

i.e.,

p^{r} ∑_{q=1}^{m} n_{q} = 2lp^{α}(p^{r−1} + p^{r−2} + ... + 1) + p^{r−1}t(p^{r−1} + ... + 1) − p(p^{r−2} + ... + 1) ∑ t_{i}

= 2lp^{α}(p^{r−1} + p^{r−2} + ... + 1) + p^{r−1}t ≡ 0 mod(p^{α+2})

Dividing both sides by p^{r−1} one gets

2lp^{α−r+2} + 2lp^{α−r+1} + t ≡ 0 mod(p^{α+3−r})

which implies t = sp^{α−r+1} and

2lp^{α+2−r} + (2l + s)p^{α−r+1} ≡ 0 mod(p^{α+3−r})
2lp + (2l + s) ≡ 0 mod(p^{2})  (3)

Now we prove if 2lp+(2l+s) ≡ 0 mod(p^{2}) then 6(g−h)−2t ≡ 0 mod(p^{α−r+2}). To reach a contradiction assume 6(g − h) − 2t ≡ 0 mod(p^{α−r+2}) and 2lp + (2l + s) ≡ 0 mod(p^{2}). Note that 2g − 2 = p^{r}(2h − 2) + p^{r−1}(p − 1)t because Γ_{g} contains π = (ℤ/p)^{r}. This implies p^{r}(2h − 2 + t) = 2g − 2 + p^{r−1}t = 2lp^{α} + p^{r−1}t = 2lp^{α} + p^{r−1}sp^{α−r+1} i.e.,

2h − 2 + t = (2l + s)p^{α−r} .

Thus,

6(g − h) − 2t = 3(2g − 2) − 3(2h − 2) − 2t = 3(2lp^{α}) − 3(2h − 2 + t) + t
= 6lp^{α} − 3(2l + s)p^{α−r} + sp^{α−r+1} ≡ 0 mod(p^{α−r+2})

which implies ps − 3(2l + s) + 6lp^{r} ≡ 0 mod(p^{2}). Note we can assume r ≥ 2, thus

ps − 3(2l + s) ≡ 0 mod(p^{2}) .

Combining this with (3) we have (6l + s)p ≡ 0 mod(p^{2}) or 6l + s ≡ 0 mod(p). It is obvious that 2l + s ≡ 0 mod(p) from (3). Thus 2s ≡ 0 mod(p) or s ≡ 0 mod(p) since p is an odd prime. Finally one has l ≡ 0 mod(p) from 2l + s ≡ 0 mod(p). This is a contradiction since we assumed g = lp^{α} + 1, l prime to p.

The next result follows immediately from Theorem 3.3 and Proposition 2.6.

Theorem 3.4. Let p be an odd prime and assume that g = lp^{α} + 1 with l prime to p and α > 0. If g satisfies the (p_{r})-condition, then p_{r}(Γ_{g}) equals 2(p − 1)p^{α−r+1}.

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