PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 127, Number 1, January 1999, Pages 297–307 S 0002-9939(99)04544-X

SMITH EQUIVALENCE OF REPRESENTATIONS FOR FINITE PERFECT GROUPS

ERKKI LAITINEN AND KRZYSZTOF PAWAŁOWSKI

(Communicated by Thomas Goodwillie)

Abstract. Using smooth one-fixed-point actions on spheres and a result due to Bob Oliver on the tangent representations at fixed points for smooth group actions on disks, we obtain a similar result for perfect group actions on spheres. For a finite group G, we compute a certain subgroup IO′(G) of the representation ring RO(G). This allows us to prove that a finite perfect group G has a smooth 2–proper action on a sphere with isolated fixed points at which the tangent representations of G are mutually nonisomorphic if and only if G contains two or more real conjugacy classes of elements not of prime power order. Moreover, by reducing group theoretical computations to number theory, for an integer n ≥ 1 and primes p, q, we prove similar results for the group G = A_{n}, SL_{2}(𝔽_{p}), or PSL_{2}(𝔽_{q}). In particular, G has Smith equivalent representations that are not isomorphic if and only if n ≥ 8, p ≥ 5, q ≥ 19.

Introduction

In 1960, P.A. Smith [Sm] asked the question: if a finite group G acts smoothly on a sphere S^{n} with exactly two fixed points x and y, it it true that T_{x}(S^{n}) ≅ T_{y}(S^{n})? The tangent spaces T_{x}(S^{n}) and T_{y}(S^{n}) are considered as the representations of G determined by the derivative of the G–action, which we refer to as to the tangent representations. In the development of transformation group theory in the subsequent decades, the isomorphism question for the tangent representations has been posed for a smooth action of G on any homotopy sphere Σ with Σ^{G} = {x, y}.

Atiyah–Bott [AB] and Milnor [Mi] answered the isomorphism question affirmatively for G = ℤ_{p}, the cyclic group of prime order p, as well as for any finite group G whose action on Σ is free outside of x and y. Without this assumption, Sanchez [Sa] proves that T_{x}(Σ) ≅ T_{y}(Σ) when G is a finite group of odd order and for each subgroup H of G, the H–fixed point set Σ^{H} is connected or Σ^{H} = {x, y}. By Smith theory, this condition on Σ^{H} holds when G is a finite p–group for a prime p, and thus, T_{x}(Σ) ≅ T_{y}(Σ) for p odd. For p = 2, there is only a weaker 2–divisibility relation between T_{x}(Σ) and T_{y}(Σ) which goes back to Bredon [Br].

In the early eighties, Cappell and Shaneson [CS1], [CS2] answered the isomorphism question negatively in the case of smooth actions of G on S^{n} with n ≥ 9 (n odd) for G = ℤ_{2^{a}q} with a ≥ 2 and q ≥ 3 (q odd), as well as in the case of smooth

\begin{matrix}c \\ ◯ \end{matrix}1999 American Mathematical Society

Received by the editors August 30, 1996 and, in revised form, May 10, 1997.

1991 Mathematics Subject Classification. Primary 57S17, 57S25.

Key words and phrases. Finite perfect group, action on sphere, Smith equivalence of representations.

actions of G on 9–dimensional homotopy spheres for G = ℤ_{4q} with q ≥ 2. It follows easily from the Atiyah–Bott–Milnor theorem and elementary representation character arguments that the smallest group G producing a negative answer to the isomorphism question is G = ℤ_{8}.

In the late seventies and early eighties, Petrie [Pe1], [Pe2], [Pe3] announced his program of developing equivariant surgery and using it to construct smooth actions on homotopy spheres. Following Petrie’s definition, we say that two representations U and V of G are Smith equivalent if there exists a smooth action of G on a homotopy sphere Σ such that Σ^{G} = {x, y} and, as representations of G, U ≅ T_{x}(Σ) and V ≅ T_{y}(Σ). By using equivariant surgery, Petrie proved that for each finite abelian group G of odd order with at least four noncyclic Sylow subgroups, there exist nonisomorphic Smith equivalent representations U and V of G (see [PR] for a detailed proof). Then, his students and collaborators obtained similar results for different classes of finite groups. In particular, the articles [DP], [DW], [DS] contain a lot of information on the order of a cyclic group G which has nonisomorphic Smith equivalent representations. We refer the reader to the survey articles [CS3], [DPS],

[MP] for more history on related results.

In this paper, a smooth action of a finite group G on a homotopy sphere Σ is called 2–proper if for each element g ∈ G of order 2^{a} with a ≥ 3, Σ^{g} is connected. If Σ^{G} = {x, y}, this amounts to saying (by Smith theory) that Σ^{g} contains Σ^{G} as a proper subset.

In accordance with Petrie’s definition, two representations U and V of G are called 2–proper Smith equivalent if there exists a smooth 2–proper action of G on a homotopy sphere Σ such that Σ^{G} = {x, y} and, as representations of G, U ≅ T_{x}(Σ) and V ≅ T_{y}(Σ).

Now, we state the main results of this paper (Theorems A and B). Hereafter, for a finite group G, we denote by r_{G} the number of real conjugacy classes of elements of G not of prime power order.

Theorem A. A finite perfect group G has nonisomorphic 2–proper Smith equivalent representations if and only if r_{G} ≥ 2. A finite perfect group G has a smooth action on a sphere with any given number k ≥ 3 of fixed points at which the tangent representations are mutually nonisomorphic if and only if r_{G} ≥ 2.

Recall that each finite simple nonabelian group is perfect. For an integer n ≥ 1, the alternating group A_{n} is simple and nonabelian (equivalently, perfect and nontrivial) if and only if n ≥ 5. Given the field 𝔽_{p} of p elements for a prime p, the special linear group SL_{2}(𝔽_{p}) is perfect if and only if p ≥ 5, and the projective special linear group PSL_{2}(𝔽_{p}) is simple (equivalently, perfect) if and only if p ≥ 5.

Theorem B. Let G = A_{n}, SL_{2}(𝔽_{p}), or PSL_{2}(𝔽_{q}) for an integer n ≥ 1 and primes p, q. Then r_{G} ≤ 1 for n ≤ 7, p ≤ 3, q ≤ 17, and r_{G} ≥ 2 for n ≥ 8, p ≥ 5, q ≥ 19. Moreover, G has nonisomorphic 2–proper Smith equivalent representations if and only if r_{G} ≥ 2. Similarly, G has nonisomorphic Smith equivalent representations if and only if r_{G} ≥ 2. Also, G has a smooth action on a sphere with any given number k ≥ 3 of fixed points at which the tangent representations are mutually nonisomorphic if and only if r_{G} ≥ 2.

This paper is divided into two parts. In Part 1, using the Deleting–Inserting Theorem [LMP, Thm. 2.2], we show that each finite nontrivial perfect group G has a smooth action on a sphere with exactly one fixed point at which the tangent representation is stably isomorphic to a given G–oriented representation V of G with dim V^{G} = 0 (Theorem 1.1). Then, using a description due to Oliver [O] of the tangent representations at fixed points for smooth finite group actions on disks (Theorem 1.2), we present a corresponding result for smooth finite perfect group actions on spheres (Theorem 1.3). Here, the idea is to take the double of the disk obtained by Oliver’s construction, and cancel the extra fixed points by forming connected sums of the doubled disk and spheres equipped with smooth one-fixed-point actions. This allows us to prove that Theorem A holds when the condition r_{G} ≥ 2 is replaced by the condition IO′(G) ≠ 0 for a certain subgroup IO′(G) of the representation ring RO(G) (Theorem 1.7). Finally, for a finite nontrivial perfect group G, we describe a subgroup of RO(G) whose elements all occur as differences of 2–proper Smith equivalent representations of G (Corollary 1.8).

In Part 2, for a finite group G, we compute the rank of the free abelian group IO′(G) in terms of the number r_{G}. In particular, IO′(G) ≠ 0 if and only if r_{G} ≥ 2 (Lemma 2.1). Theorem A now follows from Theorem 1.7. Using elementary arguments for G = A_{n} and number theory for G = SL_{2}(𝔽_{p}) or PSL_{2}(𝔽_{q}), we compute r_{G} for any n ≥ 1 and primes p, q (Propositions 2.2, 2.3, 2.4). In particular, r_{G} ≥ 2 if and only if n ≥ 8, p ≥ 5, q ≥ 19. For n ≤ 7, p ≤ 3, q ≤ 13, G has no element of order 8 (Propositions 2.2, 2.3, 2.4). With some additional arguments for PSL_{2}(𝔽_{17}) (Example 2.5 and Lemma 2.6), we show that for G = A_{n}, SL_{2}(𝔽_{p}), or PSL_{2}(𝔽_{q}), the following statement is true (Theorem 2.7). If r_{G} ≤ 1, then for any smooth action of G on a homotopy sphere, at any two isolated fixed points of the action, the tangent representations of G are isomorphic. Theorem 2.7 also contains the following two statements. If r_{G} ≥ 2, then there exist nonisomorphic 2–proper Smith equivalent representations of G. If r_{G} ≥ 2, then there exists a smooth action of G on a standard sphere with any given number k ≥ 3 of fixed points at which the tangent representations of G are mutually nonisomorphic. This proves Theorem B.

Part 1. Topology

In this paper, a real representation V of G is called G–oriented if for each subgroup H of G, the H–fixed point set V^{H} is oriented and each element of the normalizer N_{G}(H) of H in G acts on V^{H} via an orientation preserving transformation; cf. [LMP, (2.1.3)]. For a finite nonsolvable group G and a prescribed representation W of G, [LMP] constructs a smooth action of G on S^{n} (n = dim W) with exactly one fixed point x, such that T_{x}(S^{n}) ≅ W. We complement that result for a finite perfect group G, to the effect that for any real G–oriented representation V with dim V^{G} = 0 and the prescribed representation W, the action is such that T_{x}(S^{n}) ≅ V ⊕ W. This result is a special case of the main result of [LM]. However, we present a proof here to point out that, unlike in [LM], no surgery with middle-dimensional singularities is needed because we are working with the strong gap hypothesis of [LMP]. Hereafter, for a finite perfect group G, we set V (G) = ℝ[G] − ℝ, where ℝ[G] denotes the real regular representation of G and the subtracted summand ℝ has the trivial action of G.

Theorem 1.1. Let G be a finite nontrivial perfect group and V a real G–oriented representation with dim V^{G} = 0. Then, for any even integer ℓ ≥ max{6, dim V }, there exists a smooth action of G on S^{n} with exactly one fixed point x, and the action is such that T_{x}(S^{n}) ≅ V ⊕ ℓ V (G). In particular, n = dim V + ℓ (|G| − 1).

Moreover, for each proper subgroup H of G, dim(S^{n})^{H} ≥ 6 and (S^{n})^{H} is simply connected.

Proof. As G is perfect, the representation ℓ V (G) satisfies the strong gap hypothesis for the family I(G) of all proper subgroups of G, when ℓ ≥ 5 [LMP, (5.4)]. It follows that V ⊕ℓ V (G) satisfies the strong gap hypothesis when ℓ ≥ dim V [LMP, Remark to Prop. 5.1]. Moreover, for any even integer ℓ ≥ 2, ℓ V (G) is G–oriented, as it is the realification of the complex representation ℓ/2(ℂ[G] − ℂ). Now, for an even integer ℓ ≥ max{6, dim V }, set U = V ⊕ ℓ V (G). As V is G–oriented, so is U. Moreover, U satisfies the strong gap hypothesis for I(G). Let Y = S(U ⊕ ℝ) be the G–invariant sphere of U ⊕ ℝ, where G acts trivially on ℝ. Then Y^{G} = {x, y} and, as representations of G, T_{x}(Y) ≅ T_{y}(Y) ≅ U.

Now, we wish to apply the Deleting–Inserting Theorem [LMP, Thm. 2.2] to remove from Y^{G} one point, say y. In order to do it, we have to check that the assumptions (2.1.1)–(2.1.4) of Situation 2.1 [LMP] hold. Clearly, Iso(G, Y ∖ Y^{G}) = I(G). For each H ∈ I(G), Y^{H} is a sphere with dim Y^{H} ≥ 6, and hence Y^{H} is simply connected. Since U is G–oriented, thus for each H ∈ I(G), each element of N_{G}(H) acts on Y^{H} via an orientation preserving transformation. Finally, Y satisfies the strong gap hypothesis for I(G) because U does. The Deleting–Inserting Theorem [LMP, Thm. 2.2] now produces a smooth action of G on a homotopy sphere X with X^{G} = {x} and T_{x}(X) ≅ U. Moreover, for each H ∈ I(G), dim X^{H} ≥ 6 and X^{H} is simply connected. It follows from [LMP, Prop. 1.3] that X can be chosen to be the sphere S^{n} with n = dim U.

For a finite group G, consider the following subgroups of the real representation ring RO(G):

IO(G) = ⋂_{p} Ker(RO(G) → RO(G_{p})), IO′(G) = IO(G) ∩ Ker(RO(G) ⟶^{d^{G}} ℤ)

where the maps RO(G) → RO(G_{p}) are defined by restricting to Sylow subgroups G_{p} of G, and d^{G}(U − V) = dim U^{G} − dim V^{G} for representations U and V of G. Clearly, the condition that U − V ∈ IO(G) amounts to saying that U and V are isomorphic when restricted to any Sylow (and thus, any prime power order) subgroup of G. Now, we restate a theorem which goes back to Bob Oliver [O, Thm. 0.4].

Theorem 1.2. Let G be a finite group not of prime power order. Let V_{1}, …, V_{k} be real representations of G such that dim V_{i}^{G} = 0 and V_{i} − V_{j} ∈ IO(G) for all 1 ≤ i, j ≤ k. Assume k ≡ 1 (mod n_{G}). Then, for some real representation V of G with dim V^{G} = 0, there exists a smooth action of G on D^{n} with exactly k fixed points x_{1}, …, x_{k}, such that T_{x_{i}}(D^{n}) ≅ V_{i} ⊕ V for i = 1, …, k.

If G is a finite nontrivial perfect group (or more generally, a finite nonsolvable group), then the integer n_{G} = 1 (cf. [O, Thm. 0.3]). Thus, for such a group G in Theorem 1.2, there is no restriction on the number k of fixed points.

Theorem 1.3. Let G be a finite nontrivial perfect group. Let V_{1}, ..., V_{k} be real G–oriented representations such that dim V_{i}^{G} = 0 and V_{i} − V_{j} ∈ IO(G) for all 1 ≤ i, j ≤ k. Then, for some real G–oriented representation W with dim W^{G} = 0, there exists a smooth action of G on S^{n} with exactly k fixed points x_{1}, …, x_{k}, such that T_{x_{i}}(S^{n}) ≅ V_{i} ⊕ W for i = 1, …, k. Moreover, for each element g ∈ G of prime power order, (S^{n})^{g} is connected.

Proof. By Theorem 1.2, there there exist a real representation V of G with dim V^{G} = 0 and a smooth action of G on the disk D^{m} of dimension m = dim V_{i} + dim V such that D^{m} contains exactly k fixed points x_{1}, …, x_{k} at which the tangent representations are isomorphic to V_{1} ⊕ V, …, V_{k} ⊕ V. For an even integer ℓ ≥ max{6, dim(V_{i} ⊕ V ⊕ V)}, set W = V ⊕ V ⊕ ℓ V (G). Then W is a real G–oriented representation with dim W^{G} = 0.

Consider the disk D^{n} = D^{m} × D(V ⊕ ℓ V (G)) with the diagonal action of G, and form the double ∂(D^{n} × D^{1}) = S^{n} of D^{n}, where G acts trivially on D^{1}. The resulting action of G on S^{n} has exactly 2k fixed points x_{1}, y_{1}, …, x_{k}, y_{k} and, as representations of G, T_{x_{i}}(S^{n}) ≅ T_{y_{i}}(S^{n}) ≅ V_{i} ⊕W for i = 1, …, k. Since V_{i} ⊕V ⊕V is G–oriented, it follows from Theorem 1.1 that there exists a smooth action of G on a copy S_{i}^{n} of the sphere of dimension n = dim(V_{i} ⊕W) with exactly one fixed point, say z_{i}, at which the tangent representation is isomorphic to V_{i} ⊕ W. To complete the proof, we form the connected sum of S^{n} with S_{i}^{n} by piecing together a small neighborhood of y_{i} in S^{n} with a small neighborhood of z_{i} in S_{i}^{n} for i = 1, …, k. It follows from the construction that for each element g ∈ G of prime power order, dim(S^{n})^{g} ≥ 6 and, by Smith theory, (S^{n})^{g} is connected.

Lemma 1.4. Let G be a finite group acting smoothly on a homotopy sphere Σ. Assume Σ^{G} = {x, y}, and set U = T_{x}(Σ) and V = T_{y}(Σ). If the action of G on Σ is 2–proper, then U − V ∈ IO′(G).

Proof. By the Slice Theorem, dim U^{G} = dim V^{G} = 0. As recalled in the introduction, the characters χ_{U} and χ_{V} of U and V agree on elements of G of odd prime power order. They also agree on any element of order 2 or 4, because the characters of representations of cyclic groups of order 2 or 4 can be read off the fixed point set dimensions. Hence, if the action is 2–proper, χ_{U} and χ_{V} agree on all elements of prime power order, which means that U − V ∈ IO(G). Since dim U^{G} = dim V^{G}, thus U − V ∈ IO′(G).

Remark 1.5. Let G be a finite group acting smoothly on a homotopy sphere Σ. Assume Σ^{G} contains two points x and y which belong to connected components of Σ^{G} of the same dimension. Set U = T_{x}(Σ) and V = T_{y}(Σ). By the Slice Theorem, dim U^{G} = dim V^{G}. Assume further that Σ^{g} is connected for each element g ∈ G of prime power order (and thus, in particular, the action of G on Σ is 2–proper). Then, it follows that U − V ∈ IO′(G). By Smith theory, the latter assumption holds when Σ^{G} contains a third point.

Example 1.6. For a finite group G, assume IO′(G) ≠ 0 and choose U − V ≠ 0 in IO′(G). The complexifications of U − U^{G} and V − V^{G} have the realifications

U_{0} = (U − U^{G}) ⊕ (U − U^{G}) and V_{0} = (V − V^{G}) ⊕ (V − V^{G}).

As RO(G) is a free abelian group, 2(U − V) ≠ 0. In RO(G), U_{0} − V_{0} = 2(U − V). Thus, U_{0} and V_{0} are not isomorphic. For i = 1, …, k with k ≥ 2, set V_{i} = (i−1) U_{0}⊕(k−i) V_{0}. Then, each V_{i} is G–oriented, dim V_{i}^{G} = 0, and V_{i}−V_{j} ∈ IO(G). Clearly, V_{i} and V_{j} are not isomorphic when i ≠ j.

Theorem 1.7. A finite perfect group G has nonisomorphic 2–proper Smith equivalent representations if and only if IO′(G) ≠ 0. A finite perfect group G has a smooth action on a sphere with a given number k ≥ 3 of fixed points at which the tangent representations are mutually nonisomorphic if and only if IO′(G) ≠ 0.

Proof. The necessity in Theorem 1.7 follows from Lemma 1.4 and Remark 1.5. In order to prove the sufficiency, assume IO′(G) ≠ 0 and, for any integer k ≥ 2, take the mutually nonisomorphic representations V_{1}, …, V_{k} of G determined in Example 1.6. Theorem 1.3 now asserts that for some real G–oriented representation W with dim W^{G} = 0, there exists a smooth action of G on S^{n} with exactly k fixed points at which the tangent representations are isomorhic to V_{1} ⊕ W, …, V_{k} ⊕ W. In particular, the tangent representations are mutually nonisomorphic. Moreover, for each element g ∈ G of prime power order, (S^{n})^{g} is connected, and thus the action of G on S^{n} is 2–proper.

Now, for a finite nontrivial perfect group G, we describe a subgroup of RO(G) whose elements all occur as differences of 2–proper Smith equivalent representations of G. First, for a finite group G, consider the subgroups of the complex representation ring R(G):

I(G) = ⋂_{p} Ker(R(G) → R(G_{p})) and I′(G) = I(G) ∩ Ker(R(G) ⟶^{d^{G}} ℤ).

Let r : R(G) → RO(G) be the realification map. Then the following corollary holds.

Corollary 1.8. Let G be a finite nontrivial perfect group. Then each element α of the image r(I′(G)) is a difference of 2–proper Smith equivalent representations of G.

Proof. Assume α ∈ r(I′(G)). Then α = r(U − V) for two complex representations of G such that dim U^{G} = dim V^{G} and U − V ∈ I(G). The realifications U_{0} = r(U − U^{G}) and V_{0} = r(V − V^{G}) are real G–oriented representations such that dim U_{0}^{G} = dim V_{0}^{G} = 0 and U_{0} − V_{0} ∈ IO(G). Thus, by Theorem 1.3, there exists a smooth action of G on a sphere S^{n} with exactly two fixed points x and y, such that as representations of G, T_{x}(S^{n}) ≅ U_{0} ⊕ W and T_{y}(S^{n}) ≅ V_{0} ⊕ W for some real G–oriented representation W with dim W^{G} = 0. Moreover, for each element g ∈ G of prime power order, (S^{n})^{g} is connected, and thus the action of G on S^{n} is 2–proper. Clearly, in RO(G),

α = r(U − V) = r(U − U^{G}) − r(V − V^{G}) = U_{0} − V_{0} = (U_{0} ⊕ W) − (V_{0} ⊕ W).

Therefore, α is a difference of 2–proper Smith equivalent representations of G.

Part 2. Algebra

For a finite group G, we wish to compute IO′(G). As a subgroup of RO(G), the group IO′(G) is free abelian and determined by its rank rk IO′(G). Two elements g, h ∈ G are called real conjugate in G if h is conjugate to g or g^{−1}. The real conjugacy class of g is the sum (g)^{±1} := (g) ∪ (g^{−1}) of the conjugate classes of g and g^{−1}. Recall also that r_{G} denotes the number of real conjugacy classes of elements of G not of prime power order.

Lemma 2.1. Let G be a finite group. Then rk IO(G) = r_{G} and rk IO′(G) = r_{G} −1 when IO(G) ≠ 0. Otherwise, IO(G) = IO′(G) = 0. Hence, IO(G) ≠ 0 if and only if r_{G} ≥ 1, IO′(G) ≠ 0 if and only if r_{G} ≥ 2.

Proof. First, we compute the rank of IO(G) = ⋂_{p} Ker(RO(G) → RO(G_{p})). As the groups IO(G), RO(G), RO(G_{p}) are free abelian, we can tensor with ℝ to pass to the real vector spaces

ℝ ⊗_{ℤ} IO(G), ℝ ⊗_{ℤ} RO(G), ℝ ⊗_{ℤ} RO(G_{p}),

to compute rk IO(G) = dim_{ℝ}(ℝ ⊗_{ℤ} IO(G)). By [Se2, Cor. 1, p. 96], ℝ ⊗_{ℤ} RO(G) is the space of all real valued functions f on G that are constant on the real conjugacy classes (g)^{±1} for g ∈ G. Restriction to ℝ ⊗_{ℤ} RO(G_{p}) means restriction to the classes (g)^{±1} with g of p–power order. Hence, ℝ ⊗_{ℤ} IO(G) consists of the functions f which vanish on the classes (g)^{±1} with g of prime power order. Thus, rk IO(G) = dim_{ℝ}(ℝ ⊗_{ℤ} IO(G)) = r_{G}.

Now, we compute the rank of IO′(G) = Ker(IO(G) ⟶^{d^{G}} ℤ). A basis of ℝ ⊗_{ℤ} IO(G) consists of the functions f_{(g)^{±1}} which have the value 1 on (g)^{±1} and 0 otherwise, defined for all classes (g)^{±1} of elements g not of prime power order. By the fixed point set character formula (by linearity) extendable to ℝ ⊗_{ℤ} RO(G),

d^{G}(f_{(g)^{±1}}) = ⟨ f_{(g)^{±1}}, 1_{G} ⟩ = \frac{1}{|G|} ∑_{x∈G} f_{(g)^{±1}}(x) = \frac{|(g)^{±1}|}{|G|} ≠ 0.

Thus, rk IO′(G) = r_{G} − 1 when IO(G) ≠ 0. Clearly, IO(G) = 0 implies IO′(G) = 0.

Proposition 2.2. For G = A_{n}, n ≥ 1, the following two statements are true.

(1) r_{G} = 0 for n ≤ 6, r_{G} = 1 for n = 7, r_{G} ≥ 2 for n ≥ 8.

(2) For n ≤ 9, G has no element of order 8.

Proof. As each element of A_{n} has prime power order for n ≤ 6, r_{G} = 0 for n ≤ 6. For n = 7, r_{G} = 1 corresponding to the element (12)(34)(567) of order 6. For n ≥ 8, r_{G} ≥ 2 because the elements (12)(34)(567) and (123456)(78), both of order 6, are not real conjugate in G, proving (1).

Any element of A_{n} of order 8 must involve an 8–cycle in its cycle decomposition. Thus, for n ≤ 7, A_{n} has no element of order 8. Also, A_{8} and A_{9} have no elements of order 8 because an 8–cycle is not an even permutation, proving (2).

Proposition 2.3. For G = SL_{2}(𝔽_{p}), p prime, the following two statements are true.

(1) r_{G} = 0 for p = 2, r_{G} = 1 for p = 3, r_{G} ≥ 2 for p ≥ 5.

(2) For p ≤ 3, G has no element of order 8.

Proof. As SL_{2}(𝔽_{2}) ≅ S_{3}, the symmetric group on three letters, r_{G} = 0 for G = SL_{2}(𝔽_{2}). Assume p is odd and set G = SL_{2}(𝔽_{p}). Then G has three basic cyclic subgroups of orders 2p, p − 1, and p + 1 whose conjugates cover the whole group G; see [H, S. II 8.5] for the corresponding statement for PSL_{2}(𝔽_{p}). If p = 3, then p − 1 = 2, p + 1 = 4, and 2p = 6. The generators of a cyclic subgroup of G of order 6 are real conjugate in G. Thus, r_{G} = 1 for p = 3. For p ≥ 5, the numbers p − 1 and p+1 cannot both be prime powers, because then they should be powers of 2 as they are even, which would imply p = 3. This shows that G contains an element of order 2p, not a prime power, and another element not of prime power order, such that the two elements are not real conjugate in G. Thus, r_{G} ≥ 2 for p ≥ 5, proving

The Sylow 2–subgroup of G = SL_{2}(𝔽_{p}) is isomorphic to ℤ_{2} for p = 2, and the quaternion group of order 8 for p = 3. Thus, for p ≤ 3, G has no element of order 8, proving (2).

Proposition 2.4. For G = PSL_{2}(𝔽_{p}), p prime, the following two statements are true.

(1) r_{G} = 0 for p ≤ 7 and p = 17, r_{G} = 1 for p = 11 and p = 13, r_{G} ≥ 2 for p ≥ 19.

(2) For p ≤ 13, G has no element of order 8.

Proof. As PSL_{2}(𝔽_{2}) = SL_{2}(𝔽_{2}), r_{G} = 0 for G = PSL_{2}(𝔽_{2}). Assume p is odd and set G = PSL_{2}(𝔽_{p}) = SL_{2}(𝔽_{p})/{I, −I}. Similarly as in the proof of Proposition 2.3, the group G has three basic cyclic subgroups of orders p, \frac{p−1}{2}, and \frac{p+1}{2} whose conjugates cover the whole group G. The normalizers of the subgroups of orders \frac{p−1}{2} and \frac{p+1}{2} are dihedral groups of orders p−1 and p+1, respectively. This means that if we find a cyclic subgroup C of G of order not a prime power and larger than 6, then r_{G} ≥ 2 because the generators (elements not of prime power order) of C form at least two real conjugacy classes in G. For a cyclic subgroup C of G of order 6, the generators of C are real conjugate in G.

For p ≤ 7 and p = 17, \frac{p−1}{2} and \frac{p+1}{2} both are prime powers, and thus r_{G} = 0. For p = 11 and p = 13, \frac{p−1}{2} and \frac{p+1}{2} yield a prime power and 6, and thus r_{G} = 1. As we check now, for p ≥ 19, \frac{p−1}{2} and \frac{p+1}{2} cannot both be prime powers, and thus r_{G} ≥ 2. The difference of \frac{p+1}{2} and \frac{p−1}{2} is 1. If \frac{p−1}{2} and \frac{p+1}{2} both are prime powers for p ≥ 5, it follows that both are greater than or equal to 2, and one is a power of 2. Then, either p = 2^{b} + 1, b = 2^{a}, and p is a Fermat prime, or p = 2^{q} − 1, q is an odd prime, and p is a Mersenne prime [HW, p. 15]. In the Fermat prime case, \frac{p+1}{2} = 2^{2^{a}−1} +1 is divisible by 3 as 2^{a} −1 is odd, and thus \frac{p+1}{2} = 3^{m}, which implies 3^{m} − 1 = 2^{2^{a}−1}. In the Mersenne prime case, \frac{p−1}{2} = 2^{q−1} − 1 is divisible by 3 as q − 1 is even, and thus \frac{p−1}{2} = 3^{n}, which implies 3^{n} + 1 = 2^{q−1}. This is possible only for m = 1 (p = 5), m = 2 (p = 17), and n = 1 (p = 7). In fact, it follows from [Se1, Lemme, p. 32] that the order of 3 in the units of ℤ_{2^{k}} divided by ⟨ −1 ⟩ is 2^{k−2} for k ≥ 3. Hence, if 2^{k} divides 3^{m} − 1 or 3^{n} + 1, it follows that m, n ≥ 2^{k−2} unless m = n = 1, and then 3^{m} − 1 > 2^{k} unless k = 3 and m = 2, proving (1).

The Sylow 2–subgroup of G = PSL_{2}(𝔽_{p}) is isomorphic to ℤ_{2} for p = 2, ℤ_{2} ⊕ ℤ_{2} for p = 3, 5, 11, 13, and the dihedral group of order 8 for p = 7. Thus, for p ≤ 13, G has no element of order 8, proving (2).

A finite group G is called 2–proper if G has no element of order 8 or, for each element g ∈ G of order 2^{a} with a ≥ 3, dim V^{g} > 0 for any irreducible representation V of G.

Example 2.5. Let G = A_{n}, SL_{2}(𝔽_{p}), or PSL_{2}(𝔽_{q}) for an integer n ≤ 9, a prime p ≤ 3, or a prime q ≤ 17. Then G is 2–proper. In fact, except for the case G = PSL_{2}(𝔽_{17}), G has no element of order 8 by Propositions 2.2, 2.3, 2.4, and thus G is 2–proper. As the Sylow 2–subgroup of PSL_{2}(𝔽_{17}) is dihedral of order 16, PSL_{2}(𝔽_{17}) contains an element of order 8 but does not contain an element of order 2^{a} with a > 3. From the atlas of finite groups, one can see that for any irreducible representation of PSL_{2}(𝔽_{17}), each element of order 8 has positive dimensional fixed point set, and thus PSL_{2}(𝔽_{17}) is 2–proper.

Lemma 2.6. Let G be a finite 2–proper group. If G acts smoothly on a homotopy sphere Σ with Σ^{G} ≠ ∅, then the action is 2–proper. In particular, any Smith equivalent representations of G are 2–proper Smith equivalent.

Proof. If G has no element of order 8 and hence no element of order 2^{a} with a ≥ 3, then the action is 2–proper. Let g be an element of G of order 2^{a} with a ≥ 3. Choose x ∈ Σ^{G} and set U = T_{x}(Σ). By Smith theory, either Σ^{g} = {x, y} for a point y ∈ Σ, or Σ^{g} is connected. By the Slice Theorem, dim Σ^{g} = dim U^{g} ≥ dim V^{g} > 0 for any irreducible summand V of U. Thus, Σ^{g} is connected and the action is 2–proper.

Theorem 2.7. Let G = A_{n}, SL_{2}(𝔽_{p}), or PSL_{2}(𝔽_{q}) for an integer n ≥ 1 and primes p, q. If r_{G} ≤ 1, then for any smooth action of G on a homotopy sphere, at any two isolated fixed points of the action, the tangent representations of G are isomorphic. If r_{G} ≥ 2, then there exist nonisomorphic 2–proper Smith equivalent representations of G. If r_{G} ≥ 2, then there exists a smooth action of G on a standard sphere with any given number k ≥ 3 of fixed points at which the tangent representations of G are mutually nonisomorphic.

Proof. Assume r_{G} ≤ 1, which by Lemma 2.1 means that IO′(G) = 0. It follows from Propositions 2.2, 2.3, 2.4 that n ≤ 7, p ≤ 3, q ≤ 17, and by Example 2.5, G is 2–proper. For a smooth action of G on a homotopy sphere Σ with isolated points x and y of Σ^{G}, set U = T_{x}(Σ) and V = T_{y}(Σ). By Lemma 2.6, the action of G on Σ is 2–proper, and by Lemma 1.4 and Remark 1.5, U − V ∈ IO′(G) = 0; that is, U and V are isomorphic.

Assume r_{G} ≥ 2, which by Lemma 2.1 means that IO′(G) ≠ 0. Now, it follows from Propositions 2.2, 2.3, 2.4 that n ≥ 8, p ≥ 5, q ≥ 19. Thus, G is perfect, and Theorem 1.7 completes the proof.

Acknowledgements

The authors of the paper would like to thank the SFB 170 “Geometrie und Analysis” at the Universität Göttingen for the invitation and the support to attend the SFB 170 closing conference, during which the basic ideas of this paper were discussed. Erkki Laitinen would like to express thanks to Hiroshima University and Krzysztof Pawałowski would like to express thanks to Universität Heidelberg for hospitality and support during the period of completing this research. Special thanks are expressed to the referee for suggestions which improved the presentation of the material, as well as for pointing out the fact, from the atlas of finite groups, that for any irreducible representation of PSL_{2}(𝔽_{17}), each element of order 8 has positive dimensional fixed point set.

Appendix (added by Krzysztof Pawałowski)

On Saturday, August 24, 1996, Erkki Laitinen died suddenly. We were in the middle of work on the question of Smith equivalence of representations. Just a few days before his death, Erkki Laitinen posed the following conjecture for a finite Oliver group; that is, a finite group G not of prime power order with n_{G} = 1.

Laitinen Conjecture. G has nonisomorphic 2–proper Smith equivalent representations if and only if r_{G} ≥ 2.

By Theorem A, the Laitinen Conjecture holds for any finite perfect group G, and by Theorem B, the same is true for the groups A_{n}, SL_{2}(𝔽_{p}), PSL_{2}(𝔽_{q}) (recall that these groups are not perfect for n = 3, 4 and primes p, q < 5). The condition that r_{G} ≥ 2 is necessary in the Laitinen Conjecture, because Lemmas 1.4, 2.1, and 2.6 yield the following lemma.

Laitinen Lemma. For a finite group G with r_{G} ≤ 1, any 2–proper Smith equivalent representations of G are isomorphic. Moreover, for a finite 2–proper group G with r_{G} ≤ 1, any Smith equivalent representations of G are isomorphic.

Remark. [Sch, Comment (2), p. 547] contains the speculation: it is likely that finite Oliver groups G all have nonisomorphic Smith equivalent representations. As follows from our results, this is not true. In fact, A_{n} for n = 5, 6, 7, and PSL_{2}(𝔽_{q}) for q = 5, 7, 11, 13, 17, all are perfect (Oliver) 2–proper groups G with r_{G} ≤ 1, and the Laitinen Lemma asserts that any Smith equivalent representations of G are isomorphic.

References

[AB] Atiyah, M.F. and Bott,R., A Lefschetz fixed point formula for elliptic complexes: II. Applications, Ann. of Math. 88 (1968), 451–491. MR 38:731

[Br] Bredon, G., Representations at fixed points of smooth actions of compact groups, Ann. of Math. 89 (1969), 515–532. MR 39:7628

[CS1] Cappell, S.E. and Shaneson, J.L., Fixed points of periodic maps, Proc. Nat. Acad. Sci. USA 77 (1980), 5052–5054. MR 81m:57029

[CS2] Cappell, S.E. and Shaneson, J.L., Fixed points of periodic differentiable maps, Invent. Math. 68 (1982), 1–19. MR 84a:57038

[CS3] Cappell, S.E. and Shaneson, J.L., Representations at fixed points, in Group Actions on Manifolds (ed. R.Schultz), Contemp. Math. 36 (1985), 151–158. MR 86h:57038

[DP] Dovermann, K.H. and Petrie, T., Smith equivalence of representations for odd order cyclic groups, Topology 24 (1985), 283–305. MR 87d:57030

[DS] Dovermann, K.H. and Suh, D.Y., Smith equivalence for finite abelian groups, Pacific J. Math. 152 (1992), 41–78. MR 93d:57069

[DW] Dovermann, K.H. and Washington, L.D., Relations between cyclotomic units and Smith equivalence of representations, Topology 28 (1989), 81–89. MR 90e:57071

[DPS] Dovermann, K.H., Petrie, T. and Schultz, R., Transformation groups and fixed point data, in Group Actions on Manifolds (ed. R. Schultz), Contemp. Math. 36 (1985), 159–189. MR 86e:57039

[HW] Hardy, G. H. and Wright, E. M., An Introduction to the Theory of Numbers, 3rd Ed., Oxford University Press, London, 1954. MR 16:673c

[H] Huppert, B., Endliche Gruppen I, Grundlehren 134, Springer, Berlin, 1967. MR 37:302 [LM] Laitinen, E. and Morimoto, M., Finite groups with smooth one fixed point actions on

spheres, Forum Math. 10 (1998), 479–520.

[LMP] Laitinen, E., Morimoto, M. and Pawałowski, K., Deleting–Inserting Theorem for smooth actions of finite nonsolvable groups on spheres, Comment. Math. Helv. 70 (1995), 10–38. MR 96b:57043

[MP] Masuda, M. and Petrie, T., Lectures on transformation groups and Smith equivalence, in Group Actions and Manifolds (ed. R. Schultz), Contemp. Math. 36 (1985), 191–242. MR 87a:57047

[Mi] Milnor, J.W., Whitehead torsion, Bull. Amer. Math. Soc. 72 (1966), 358–426. MR 33:4922 [O] Oliver, B., Fixed point sets and tangent bundles of actions on disks and Euclidean spaces,

Topology 35 (1996), 583–615. MR 97g:57059

[Pe1] Petrie, T., G–surgery I — A survey, in Algebraic and Geometric Topology, Proceedings (Santa Barbara 1977), Lecture Notes in Math. 664 (1978), 197–233. MR 80g:57049

[Pe2] Petrie, T., Pseudoequivalences of G–manifolds, in Algebraic and Geometric Topology, Proc. Symp. in Pure Math. 32 (1978), 169–210. MR 80e:57039

[Pe3] Petrie, T., Smith equivalence of representations, Math. Proc. Cambridge Philos. Soc. 94 (1983), 61–99. MR 85i:57011

[PR] Petrie, T. and Randall, J., Spherical isotropy representations, Publ. Math. IHES 62 (1985), 5–40. MR 88i:57017 [Sa] Sanchez, C.U., Actions of groups of odd order on compact orientable manifolds, Proc.

Amer. Math. Soc. 54 (1976), 445–448. MR 53:11641

[Sch] Schultz, R., Problems submitted to the A.M.S. Summer Research Conference on Group Actions, Collected and edited by R. Schultz, in Group Actions on Manifolds, Contemp. Math., vol. 36, 1985, pp. 513–568. MR 86i:57042

[Se1] Serre, J.–P., Cours d’Arithmétique, Presses Universitaires de France, Paris, 1970. MR 41:138

[Se2] Serre, J.–P., Linear Representations of Finite Groups, GTM 42, Springer, New York, 1977. MR 56:8675

[Sm] Smith, P.A., New results and old problems in finite transformation groups, Bull. Amer. Math. Soc. 66 (1960), 401–415. MR 23:A2880

Faculty of Mathematics and Computer Science, Adam Mickiewicz University of Poznań, ul. Jana Matejki 48/49, PL–60–769 Poznań, Poland

   E-mail address: kpa@math.amu.edu.pl