By Jonah Robotham

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The map G · (x1 , g · y1 ) → HgU deﬁnes the second one-to-one correspondence. 12 If O is a G-orbit on X × Y and (x, y) ∈ O then O(x) = StabG (x) · y and O (y) = StabG (y) · x. Hence |O| = [G : StabG ((x, y))] = [G : StabG (x)]|O(x)| = [G : StabG (y)]|O (y)|. Hence, if Ω = X = Y is a transitive G-set, then |O(x)| = |O (y)| = |O| |Ω| for arbitrary (x, y) ∈ O. 11 is the same as the number of orbits of H = StabG (ω) on Ω for any ω ∈ Ω) is called the rank of Ω and also of KΩ. If the rank is 2 then one also says that G acts doubly transitively on Ω.

1 Let Ω be a ﬁnite non-empty set. Note that Ω or more precisely (Ω, ·) is called a (left) G-set, and G is said to act on Ω (from the left) if ·: G × Ω → Ω (g, ω) → g · ω is a map satisfying g1 · (g2 · ω) = (g1 g2 ) · ω 1G · ω = ω for all g1 , g2 ∈ G, ω ∈ Ω, for all ω ∈ Ω. 11) If Ω1 , Ω2 are G-sets then a map ϕ : Ω1 → Ω2 is called G-equivariant or a G-map if ϕ(g · ω) = g · ϕ(ω) for all g ∈ G , ω ∈ Ω1 . The set of G-maps from Ω1 to Ω2 is usually denoted by HomG (Ω1 , Ω2 ). If, in addition, ϕ is bijective then ϕ is called a G-isomorphism, and if such a ϕ exists Ω1 , Ω2 are called isomorphic G-sets; in symbols, Ω1 ∼ =G Ω2 .

6). There is a variant of the Meataxe algorithm due to Holt and Rees [86], which has a performance quite independent of the size of the ﬁnite ﬁeld K that we are going to present. It is based on the following lemma. 9 Let A be an algebra over a ﬁeld K and V be an A-module of ﬁnite dimension over K. Let f be an irreducible factor of the characteristic polynomial ca of δV (a) for some a ∈ A. Assume that deg f = dimK kerV (f (a)). Then V is a simple A-module if (and only if ) (a) A · v = V for some 0 = v ∈ kerV (f (a)) and (b) x · A = V for some 0 = x ∈ kerV (f (a)).