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

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  and . In , 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 , 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 .

References

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

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

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

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

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

  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.