The Laitinen Conjecture (posed by Erkki Laitinen in 1996) predicts that the answer to the Smith question is negative in the case where G is a finite Oliver group with two or more real conjugate classes of elements not of prime power order, i.e., with rc(G) ≥ 2. The Laitinen Conjecture asserts also that the related action of G satisfies the 8-condition. As pointed out at the Smith Problem page, the condition that rc(G) ≥ 2 is necessary for the existence of such an action of G.

The Laitinen Conjecture is verified in many cases and my contribution reads as follows.

Under each of the condition on G below, there exists a smooth action of G on a sphere with exactly two fixed points, satisfying the 8-condition, at which the tangent G-modules are not isomorphic.

  • G is a finite perfect group with rc(G) ≥ 2 (see [1]).
  • G is a finite Oliver group of odd order. Then rc(G) ≥ 2 (see [2]).
  • G is a finite Oliver group with quotient C_{pq} for (p,q) =1, p,q odd. Then rc(G) ≥ 2 (see [2]).
  • G is a finite non-solvable gap group with rc(G) ≥ 2, G ≠ PΣL(2,27) (see [2]).
  • G is a finite non-solvable group with rc(G) ≥ 2, G ≠ PΣL(2,27) and G ≠ Aut(A_6) (see [3]).

Here, Aut(A_6) is the group of automorphisms of the alternating group A_6 of even permutations on 6 elements. Moreover, PΣL(2,27) is an extension of the projective special linear group PSL(2,27) by the group of automorphism of the filed of 27 elements, acting on PSL(2,27) in the canonical way.

For G = PΣL(2,27) or Aut(A_6), rc(G) = 2. The two exceptional cases of finite non-solvable groups are covered by Morimoto [4] and [5]. In [4], he has proved that for = Aut(A_6), the answer to the Smith question is affirmative and thus, the Laitinen Conjecture is not true. In turn, in [5], he has proved that for G = PΣL(2,27), the Laitinen Conjecture is true. Therefore, the Laitinen Conjecture is true for all but one finite non-solvable groups with rc(G) ≥ 2.

A finite group G is nilpotent if and only if G is the product of its Sylow subgroups, and a finite nilpotent group G is an Oliver group if and only if G has at least three distinct Sylow subgroups. Therefore, if G is a finite nilpotent Oliver group, then G has a quotient C_{pq} for (p,q) =1, p,q odd.

A survey of results (obtained up to 2017) related to the Smith Problem and the Laitinen Conjecture is given in [6].

References

[1]  Laitinen, E.; Pawałowski, K, Smith equivalence of representations for finite perfect groups,
Proc. Am. Math. Soc. 127 (1999), 297–307.

[2]  Pawałowski, K.; Solomon, R., Smith equivalence and finite Oliver groups,
Algebraic & Geometric Topology 2 (2002), 843–895.

[3]  Pawałowski, K.; Sumi, T., The Laitinen Conjecture for finite non-solvable groups,
Proc. Edinburgh Math. Soc. 56 (2013), 303–336.

[4]  Morimoto, M., Smith equivalent Aut(A_6)-representations are isomorphic,
Proc.Amer. Math. Soc. 136 (2008), 3683–3688.

[5]  Morimoto, M., Nontrivial P(G)-matched S-related pairs for finite gap Oliver groups,
J. Math. Soc. Japan 62, no. 2, (2010), 623–647.

[6]  Pawałowski, K. M., The Smith Equivalence Problem and the Laitinen Conjecture,
Handbook of Group Actions, Vol. III, ALM 40, International Press of Boston (2018), 485–538.