Pronormality and Submaximal $$\mathfrak {X}$$ X -Subgroups on Finite Groups

The efficient degradation of organic pollutants in an aqueous environment under visible light irradiation by persulfate catalytically activated with kaolin-Fe2O3

“…As we know, for the same oxidation system, the degradation effect may vary greatly with different concentrations of pollutant. 31 The inuence of the concentration of RhB in the initial solution on the degradation of RhB in the K-Fe 2 O 3 /PS/visible light system is shown in Fig. 6c.…”

An Example of a Relatively Maximal Nonpronormal Subgroup of Odd Order in a Finite Simple Group

“…In 2018 in [4,Problem 5.20] and [7,Problem 6], the problem was posed even more widely, as the problem of the pronormality of the so-called X-submaximal subgroups which include X-maximal subgroups. The works [7][8][9] and the survey [4] can give an impression about X-submaximal subgroups and the role of the pronormality problems in their study, especially for the groups that are simple or Original article submitted December 6, 2023; revised December 6, 2023; accepted January 25, 2024. close to simple.…”

“…In 2018 in [4,Problem 5.20] and [7,Problem 6], the problem was posed even more widely, as the problem of the pronormality of the so-called X-submaximal subgroups which include X-maximal subgroups. The works [7][8][9] and the survey [4] can give an impression about X-submaximal subgroups and the role of the pronormality problems in their study, especially for the groups that are simple or Original article submitted December 6, 2023; revised December 6, 2023; accepted January 25, 2024. close to simple. An additional reason for studying Problem 1 is the pronormality of Hall subgroups (proved in [10]), i.e., of subgroups for which the index and the order are coprime; or, equivalently, of X-Hall subgroups in simple groups, i.e., of X-subgroups whose indices do not divide by primes p if only X contains a group of order p. Obviously, X-Hall subgroups are always X-maximal.…”

“…In spite of the negative solution of Problem 1, advances in studying the problem of the pronormality of subgroups of odd index (see [3,4]) give a reason to hope that at least maximal X-subgroups of odd index in finite simple groups are always pronormal [11,Problem 2].…”

“…There is an extensive literature on the problems related to this property (see the surveys [2][3][4]).…”