M.D. Pérez-Ramos’s research while affiliated with University of Valencia and other places

... According to [8, Proposition 3.1], 2-generated connected involutory quandles are affine. Let Q be the involutory quandle SmallQuandle (15,6) of the RIG database. The quandle Q is latin and so its 2-generated subquandles are affine. ...

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On Core Quandles

... Evan [8] and Brewster et al. [6] provided some sufficient and necessary conditions for a diagonal-type subgroup of G × G to be a permutable subgroup. In the survey article [5], the authors investigated various embeddings of subgroups in direct products. In [5,Theorem 3.3], the authors using results from [8,11] proved results similar to those in this paper under different normality conditions. ...

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How does Diagonal Subgroup Embedding Determine the Structure of a Group?

... Structure and properties of N -connected products, for the class N of finite nilpotent groups, are well known (cf. [1,14,2]); for instance, G = AB is an N -connected product of A and B if and only if G modulo its hypercenter is a direct product of the images of A and B. Apart from the above-mentioned results regarding S-connection, corresponding studies for the classes N 2 and N A of metanilpotent groups, and groups with nilpotent derived subgroup, respectively, have been carried out in [8,9]; in [10] connected products for the class S π S ρ of finite soluble groups that are extensions of a normal π-subgroup by a ρ-subgroup, for arbitrary sets of primes π and ρ, are studied. The class S π S ρ appears in that reference as the relevant case of a large family of formations, named nilpotent-like Fitting formations, which comprise a variety of classes of groups, such as the class of π-closed soluble groups, or groups with Sylow towers with respect to total orderings of the primes. ...

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Products of finite connected subgroups

... They have obtained the following nice result: Assume that G = AB is the product of two supersoluble subgroups A and B. If every subgroup of A is permutable with every subgroup of B, then G is supersoluble. In addition, they have also generalized the above mentioned result of Baer by replacing the condition of normality of A, B in G and using the following weaker condition: A permutes with all subgroups of B and B permutes with all subgroups of A. Their results in [3] were further developed and applied by many authors (see, for example, [1] [5][6][7][8], [14], [19]). We also notice that O.H.Kegel has also obtained many elegant results for soluble groups and supersoluble groups by considering the products of their subgroups (see [16][17][18]). ...

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Criterions of Supersolubility for Products of Supersoluble Groups

... Consequently, applying the main result of [3] (see [1, Theorem 6.2.19]), S G ðX, rÞ ¼ hg 2 G : g 2 XhX, X g i à i for every subgroup X of a soluble group G. ...

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On σ -subnormal closure

... Then, inspired by the previous research on totally permutable products of subgroups, an initial study on conditional permutability in the framework of formation theory has been developed in [3]. A compilation of recent results can be found in [2]. Easy examples in the previous references ([1, Examples 2, 3]; also [3, Examples 3.5, 3.6]) show that strong structural properties of products of totally permutable subgroups are missed when permutability is weakened to conditional permutability, even complete, and make evident the interest of the recent progress. ...

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On conditional permutability and factorized groups

... Lemma 4 [15,Lemma 6] Let G be the symmetric group of degree k and let s be a prime. If s N is the largest power of s dividing |G| = k!, then N ≤ k−1 s−1 . ...

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On products of groups and indices not divisible by a given prime

... Focusing in products of groups, along the last decades, some relations of permutability between the factors have been considered by many authors, as, for instance, total permutability, mutual permutability (see [4]) and tcc-permutability (see [1,2]). These last permutability relations are inherited by quotients, and they ensure the existence of a minimal normal subgroup contained in one of the factors. ...

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Zeros of irreducible characters in factorised groups

... The local version of the above question is the one appearing in The Kourovka Notebook ([MK90]) as Question 9.58. It was posed by Shemetkov and Skiba and answered affirmatively in several papers (see [BBPR98,Ved88,Vor93]). ...

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On partially saturated formations of finite groups

... Note that the product of Hall π -subgroups of A and B, respectively, need not be a π -subgroup of G in this case. The proof uses theorems in [67] and the results in [9] on the ranks of the products of two metacyclic groups. ...

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Factorizations of Finite Groups and Related Topics