An identification theorem for the sporadic simple groups F2 and M(23)

We identify the sporadic groups M(23) and F

AN IDENTIFICATION THEOREM FOR GROUPS WITH SOCLE PSU

We identify the groups ${\text{PSU} }_{6} (2)$, ${\text{PSU} }_{6} (2){: }2$, ${\text{PSU} }_{6} (2){: }3$ and $\text{Aut} ({\text{PSU} }_{6} (2))$ from the structure of the centralizer of an element of order three.

Actions of Groups of Finite Morley Rank on Small Abelian Groups

We classify actions of groups of finite Morley rank on abelian groups of Morley rank 2: there are essentially two, namely the natural actions of SL(V) and GL(V) with V a vector space of dimension 2. We also prove an identification theorem for the natural module of SL2 in the finite Morley rank category.

DUAL GEOMETRY OF THE WIGNER–YANASE–DYSON INFORMATION CONTENT

We develop a non-parametric information geometry on finite-dimensional matrix manifolds by using the Fréchet differentiation. Taking the simplest prototype Riemannian metric form [Formula: see text], [Formula: see text] with the Fréchet derivative D on a pair of smooth functions g(ρ) and g*(ρ) of the density matrix ρ, we prove the WYD identification theorem: this metric is identified with the normalized Wigner–Yanase–Dyson skew information (the WYD metric), if and only if the metric form satisfies the monotonicity under every stochastic mapping T on ρ and A: [Formula: see text]. On this basis, we establish (a) a fine structure of the partial order in the set ℱ of all monotone metrics such that ℱ power ∪ℱ WYD forms a linearly ordered subset of ℱ with the same mini-max bound, where ℱ power (the power-mean metrics) interpolates the Bures and the WYD metrics (b) a characterization of the quasi-entropy S (ρ, σ)= Tr F (Δσ, ρ)ρ induced by the metric-characterizing function f WYD (x) of the WYD metric (c) an affine connection on the above metric which is torsionless to guarantee the quantum version of the ±α-connection, provided α∈[-3, 3].