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The A-Möbius function of a finite group
Full text linkThe Möbius function of the subgroup lattice of a finite group G has been introduced by Hall and applied to investigate several different questions. We propose the following generalization. Let A be a subgroup of the automorphism group Aut(G) of a finite group G and denote by CA(G) the set of A-conjugacy classes of subgroups of G. For H ≤ G let [H]A = { Ha | a ∈ A} be the element of CA(G) containing H. We may define an ordering in CA(G) in the following way: [H]A ≤ [K]A if Ha ≤ K for some a ∈ A. We consider the Möbius function μA of the corresponding poset and analyse its properties and possible applications
Minimal counterexamples to a conjecture of Hall and Paige
No full textA complete map for a group G is a permutation phi: G --> G such that g bar right arrow g phi (g) is still a permutation of G. A conjecture of M. Hall and L. J. Paige states that every finite group whose Sylow 2-subgroup is non-trivial and non-cyclic admits a complete map. In the present paper it is proved that a potential counterexample G of minimal order to this conjecture either is almost simple or G has only one involution, the Sylow 2-subgroups of G are quaternionic, \G/G\ less than or equal to 2, G' congruent to SL(2, q) for some odd prime power q > 5 and if G is not a perfect group then G/Z(G') congruent to PGL(2, q)
Finite groups that need more generators than any proper quotient
No full textA structure theorem is proved for finite groups with the property that, for some integer m with m ≥ 2, every proper quotient group can be generated by m elements but the group itself cannot
On two Möbius functions for a finite non-solvable group
No full textLet G be a finite group, μ be the Möbius function on the subgroup lattice of G, and λ be the Möbius function on the poset of conjugacy classes of subgroups of G. It was proved by Pahlings that, whenever G is solvable, the property (Formula presented.) holds for any subgroup H of G. It is known that this property does not hold in general, the Mathieu group M 12 being a counterexample. In this paper we investigate the relation between μ and λ for some classes of non-solvable groups, among them, the minimal non-solvable groups. We also provide several examples of groups not satisfying the property
On the classification problem for the genera of quotients of the Hermitian curve
Full text linkIn this article, we characterize the genera of those quotient curves H q /G of the F q 2-maximal Hermitian curve H q for which either G is contained in the maximal subgroup M 1 of (H q ) fixing a self-polar triangle, or q is even and G is contained in the maximal subgroup M 2 of (H q ) fixing a pole-polar pair (P,l) with respect to the unitary polarity associated to H q (F q 2) In this way, several new values for the genus of a maximal curve over a finite field are obtained. Our results leave just two open cases to provide the complete list of genera of Galois subcovers of the Hermitian curve; namely, the open cases in [4] when G fixes a point P∈H q (F q 2) and q is even, and the open cases in [33] when G≤M 2 and q is odd
The Mobius function of PSL (3, 2 p) for any prime p
No full textLet G be the simple group PSL(3, 2p), where p is a prime number. For any subgroup H of G, we compute the Mobius function μ(H) of H in the subgroup lattice of G. To this aim, we describe the intersections of maximal subgroups of G. We point out some connections of the Mobius function with other combinatorial objects, and, in this context, we compute the reduced Euler characteristic of the order complex of the subposet of r-subgroups of PGL(3,q), for any prime r and any prime power q
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