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Subgroups of finite solvable groups inducing the same permutation character
No full textIn this paper there are found necessary and sufficient conditions that a pair of solvable finite groups, say G and K, must satisfy for the existence of a solvable finite group L containing two isomorphic copies of G and H inducing the same permutation character. Also a construction of L is given as an iterated wreath product, with respect to their actions on their natural modules, of finite one-dimensional affine groups
Ideally constrained Lie algebras
No full textIn this paper we deal with graded Lie algebras L such that there exists a positive integer r such that for every positive integer i and for every homogeneous ideal I ⊈ L i the inclusion I ⊇ L i+r-1 holds. The solvable case and the r = 1 case receive a special attention
Metabelian thin Beauville p-groups
No full textA non-cyclic finite p-group G is said to be thin if every normal subgroup of G lies between two consecutive terms of the lower central series and |γi(G):γi+1(G)|≤p2 for all i≥1. In this paper, we determine Beauville structures in metabelian thin p-groups
Complete mappings in some linear and projective groups
No full textIn this paper we prove that, in even characteristic, the groups , , and ,
admit a complete mapping, with the exception of . Moreover, the same is proved for the Mathieu groups,their automorphism groups and ,
On the admissibility of some linear and projective groups in odd characteristic
No full textA bijective mapping phi: G --> G defined on a finite group G is complete if the mapping eta defined by eta(x) = x phi(x), x is an element of G, is bijective. In 1955 M. Hall and L. J. Paige conjectured that a finite group G has a complete mapping if and only if a Sylow 2-subgroup of G is non-cyclic or trivial. This conjecture is still open. In this paper we construct a complete mapping for the projective groups PSL(2, q), q = 1 mod 4 and PGL(2, q), q odd. As a consequence, we prove that in odd characteristic the projective groups PGL(n, q), n > 2, and the linear groups GL(n, q), n greater than or equal to 2, admit a complete mapping
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)
Metabelian thin Lie algebras
No full textA graded Lie algebra is thin if it is generated by two elements of degree 1 and each of its homogeneous ideals is located between two consecutive terms of the lower central series. In this paper we give a complete classification of the metabelian thin Lie algebras and their graded automorphism groups
A modular idealizer chain and unrefinability of partitions with repeated parts
Full text linkRecently Aragona et al. have introduced a chain of normalizers in a Sylow 2-subgroup of Sym(2^n), starting from an elementary abelian regular subgroup. They have shown that the indices of consecutive groups in the chain depend on the number of partitions into distinct parts and have given a description, by means of rigid commutators, of the first n − 2 terms in the chain. Moreover, they proved that the (n − 1)-th term of the chain is described by means of rigid commutators corresponding to unrefinable partitions into distinct parts. Although the mentioned chain can be defined in a Sylow p-subgroup of Sym(p^n), for p > 2 computing
the chain of normalizers becomes a challenging task, in the absence of a suitable notion of rigid commutators. This problem is addressed here from an alternative point of view. We propose a more general framework for the normalizer chain, defining a chain of idealizers in a Lie ring over Z_m whose elements are represented by integer partitions. We show how the corresponding idealizers are generated by subsets of partitions into at most m − 1 parts and we conjecture that the idealizer chain grows as the normalizer chain in the symmetric group. As evidence of this, we establish a correspondence between the two constructions in the case m = 2
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