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 *G *= 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 *G *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**

*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.