Topological complexity and L-S theory

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Sakai and Iwase showed the folloing. 1) A space X with cat(X) less than or equal to k is an H-space if and only if X is a C_k soace. Moreover, a C_1 space with at most three cells other than the base point is an H-space. 2) A monoidal topological complexity differs from the original topological complexity at most by 1. 3) The L-S category of a rotation group coincides with its cup-length. 4) L-S category is less than or equal to 1 if and only if p-localized version of L-S category has the same property for every prime p, where we do not need to assume that the space is simply-connected. Further Sumi et al gave 5) a necessary and sufficient condition for a finite group G to be a gap group, unless the first homology group H_1(G) is a 2-group and showed 6) The Laitinen conjecture is true for any finite non-solvable groups except for Aut(A_6).