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39 (2), Γ is L-invariant so, like ∆, it is an H-invariant set of blocks for the action of H on I. Thus if ∆ is a minimal such partition of I, then H is maximal in G. Finally, any H-invariant block is G-invariant, by 6. 24. If the projection of U ∩ Soc(G) on each component Si of Soc(G) is surjective, then U ∩ Soc(G) = D1 × · · · × Dl , with 1 ≤ l < n, and each Di is isomorphic to S. Hence Soc(G) = U ∩ Soc(G) K1 and then G = U K1 . 25. In this study we have observed three diﬀerent types of core-free maximal subgroups U of a primitive group G of type 2 according to the image of the projection π1 : U ∩ Soc(G) −→ S1 .

If N is a normal subgroup of G and K ≤ N ≤ H, then either H = N or K = N . Equivalently, H/K is a chief factor of G if H/K is a minimal normal subgroup of G/K. Hence H/K is a direct product of copies of a simple group and we have two possibilities: 1. either H/K is abelian, and there exists a prime p such that H/K is an elementary abelian p-group, or 2. H/K is non-abelian, and there exists a non-abelian simple group S such that H/K ∼ = S for all i = 1, . . , n. = S1 × · · · × Sn , where Si ∼ Given a group G and two normal subgroups K, H of G such that K ≤ H, the group G acts by conjugation on the cosets of the section H/K: for h ∈ H and g ∈ G, then (hK)g = hg K.

14. Since H2 C/C is also a minimal normal subgroup of G/C, then H1 C = H2 C. Hence H1 /K1 and H2 /K2 are G-isomorphic. To see that this does not hold when the chief factors are abelian, let P be an extraspecial p-group, p an odd prime, of order p3 . Let F be a ﬁeld of characteristic q, with q = p, such that F contains a primitive p-th root of unity. 16]). Since p − 1 > 1, we can consider two non-isomorphic such P -modules, V1 , V2 . If V is the direct sum V = V1 ⊕ V2 , construct the semidirect product G = [V ]P .