The index complex of a maximal subalgebra of a Lie algebra.

Let M be a maximal subalgebra of the Lie algebra L. A subalgebra C of L is said to be a completion for M if C is not contained in M but every proper subalgebra of C that is an ideal of L is contained in M. The set I(M) of all completions of M is called the index complex of M in L. We use this concept to investigate the influence of the maximal subalgebras on the structure of a Lie algebra, in particular finding new characterisations of solvable and supersolvable Lie algebras

A note on supersoluble maximal subgroups and Theta-pairs

A 0-pair for a maximal subgroup M of agroup G is a pair (A, B) of subgroups such that B is a maximal G-invariant subgroup of A with B but not A contained in M. 0-pairs are considered here in some groups having supersoluble maximal subgroups

Some local properties defining T0T_0-groups and related classes of groups

We call G a Hall_X -group if there exists a normal nilpotent subgroup N of G for which G/N' is an X -group. We call G a T_0-group provided G/Φ(G) is a T -group, that is, one in which normality is a transitive relation. We present several new local classes of groups which locally define HallX -groups and T0 -groups where X ∈ {T , PT , PST }; the classes PT and PST denote, respectively, the classes of groups in which permutability and S-permutability are transitive relation

Transitivity of Sylow permutability, the converse of Lagrange's theorem, and mutually permutable products

This paper is devoted to the study of mutually permutable products of finite groups. A factorised group G = AB is said to be a mutually permutable product of its factors A and B when each factor permutes with every subgroup of the other factor. We prove that mutually permutable products of Y -groups (groups satisfying the converse of Lagrange's theorem) and SC-groups (groups whose chief factors are simple) are SC -groups. Next, we show that a product of pairwise mutually permutable Y -groups is supersoluble. Finally, we give a local version of the result stating that if a mutually permutable product of two groups is a PST - group (that is, a group in which every subnormal subgroup permutes with all Sylow subgroups), then both factors are PST -group

Permutable subnormal subgroups of finite groups

The aim of this paper is to prove certain characterization theorems for groups in which permutability is a transitive relation, the so called PT -groups. In particular, it is shown that the finite solvable PT -groups, the finite solvable groups in which every subnormal subgroup of defect two is permutable, the finite solvable groups in which every normal subgroup is permutable sensitive, and the finite solvable groups in which conjugate-permutability and permutability coincide are all one and the same class. This follows from our main result which says that the finite modular p-groups, p a prime, are those p-groups in which every subnormal subgroup of defect two is permutable or, equivalently, in which every normal subgroup is permutable sensitive. However, there exist finite insolvable groups which are not PT -groups but all subnormal subgroups of defect two are permutable

Maximal subgroups and PST-groups

A subgroup H of a group G is said to permute with a subgroup K of G if HK is a subgroup of G. H is said to be permutable (resp. S-permutable) if it permutes with all the subgroups (resp. Sylow subgroups) of G. Finite groups in which permutability (resp. S-permutability) is a transitive relation are called PT-groups (resp. PST-groups). PT-, PST- and T-groups, or groups in which normality is transitive, have been extensively studied and characterised. Kaplan [Kaplan G., On T-groups, supersolvable groups, and maximal subgroups, Arch. Math. (Basel), 2011, 96(1), 19-25] presented some new characterisations of soluble T-groups. The main goal of this paper is to establish PT- and PST-versiosn of Kaplan's results, which enables a better understanding of the relationships between these classes

On a class of supersoluble groups

A subgroup H of a finite group G is said to be S-semipermutable in G if H permutes with every Sylow q-subgroup of G for all primes q not dividing |H|. A finite group G is an MS-group if the maximal subgroups of all the Sylow subgroups of G are S-semipermutable in G. The aim of the present paper is to characterise the finite MS-groups

Cross acclimation between heat and hypoxia: Heat acclimation improves cellular tolerance and exercise performance in acute normobaric hypoxia

Background: The potential for cross acclimation between environmental stressors is not well understood. Thus, the aim of this investigation was to determine the effect of fixed-workload heat or hypoxic acclimation on cellular, physiological, and performance responses during post acclimation hypoxic exercise in humans. Method: Twenty-one males (age 22 ± 5 years; stature 1.76 ± 0.07 m; mass 71.8 ± 7.9 kg; V˙O2 peak 51 ± 7 mL.kg−1.min−1) completed a cycling hypoxic stress test (HST) and self-paced 16.1 km time trial (TT) before (HST1, TT1), and after (HST2, TT2) a series of 10 daily 60 min training sessions (50% N V˙O2 peak) in control (CON, n = 7; 18°C, 35% RH), hypoxic (HYP, n = 7; fraction of inspired oxygen = 0.14, 18°C, 35% RH), or hot (HOT, n = 7; 40°C, 25% RH) conditions. Results: TT performance in hypoxia was improved following both acclimation treatments, HYP (−3:16 ± 3:10 min:s; p = 0.0006) and HOT (−2:02 ± 1:02 min:s; p = 0.005), but unchanged after CON (+0:31 ± 1:42 min:s). Resting monocyte heat shock protein 72 (mHSP72) increased prior to HST2 in HOT (62 ± 46%) and HYP (58 ± 52%), but was unchanged after CON (9 ± 46%), leading to an attenuated mHSP72 response to hypoxic exercise in HOT and HYP HST2 compared to HST1 (p < 0.01). Changes in extracellular hypoxia-inducible factor 1-α followed a similar pattern to those of mHSP72. Physiological strain index (PSI) was attenuated in HOT (HST1 = 4.12 ± 0.58, HST2 = 3.60 ± 0.42; p = 0.007) as a result of a reduced HR (HST1 = 140 ± 14 b.min−1; HST2 131 ± 9 b.min−1 p = 0.0006) and Trectal (HST1 = 37.55 ± 0.18°C; HST2 37.45 ± 0.14°C; p = 0.018) during exercise. Whereas PSI did not change in HYP (HST1 = 4.82 ± 0.64, HST2 4.83 ± 0.63). Conclusion: Heat acclimation improved cellular and systemic physiological tolerance to steady state exercise in moderate hypoxia. Additionally we show, for the first time, that heat acclimation improved cycling time trial performance to a magnitude similar to that achieved by hypoxic acclimation