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For a finite group G and a fixed Sylow p-subgroup P of G, Ballester-Bolinches and Guo proved in 2000 that G is p-nilpotent if every element of P ∩ G′ with order p lies in the center of NG(P) and when p = 2, either every element of P ∩ G′ with order 4 lies in the center of NG(P) or P is quaternion-free and NG(P) is 2-nilpotent. Asaad introduced weakly pronormal subgroup of G in 2014 and proved that G is p-nilpotent if every element of P with order p is weakly pronormal in G and when p = 2, every element of P with order 4 is also weakly pronormal in G. These results generalized famous Itô's Lemma. We are motivated to generalize Ballester-Bolinches and Guo's Theorem and Asaad's Theorem. It is proved that if p is the smallest prime dividing the order of a group G and P, a Sylow p-subgroup of G, then G is p-nilpotent if G is S4-free and every subgroup of order p in P ∩ Px ∩ G N p is weakly pronormal in NG(P) for all x ∈ GNG(P), and when p = 2, P is quaternion-free, where G N p is the p-nilpotent residual of G. [ABSTRACT FROM AUTHOR]

A subgroup H of a finite group G is said to be nearly s-embedded in G if G has an s-permutable subgroup T and an s-semipermutable subgroup HssG contained in H such that H s G = H T and H ∩ T ≤ H ssG , where HsG is the intersection of all s-permutable subgroups of G containing H. In this article, we investigate the influence of some nearly s-embedded subgroups of Sylow p-subgroups on the p-nilpotency of finite groups. A number of known results are improved and extended. [ABSTRACT FROM AUTHOR]

Throughout this article, all groups are finite and G is a group. Let σ = { σ i | i ∈ I } be some partition of the set of all primes P. Then σ (G) = { σ i | σ i ∩ π (G) ≠ ∅ } ; σ + (G) = { σ i | G has a chief factor H/K such that σ (H / K) = { σ i } }. The group G is said to be: σ-primary if G is a σi-group for some i; σ-soluble if every chief factor of G is σ-primary. The symbol R σ (G) denotes the product of all normal σ-soluble subgroups of G. The chief factor H/K of G is said to be: σ-central (in G) if (H / K) ⋊ (G / C G (H / K)) is σ-primary; a σi-factor if H/K is a σi-group. We say that G is: σ-nilpotent if every chief factor of G is σ-central; generalized { σ i } -nilpotent if every chief σi-factor of G is σ-central. The symbol F { g σ i } (G) denotes the product of all normal generalized { σ i } -nilpotent subgroups of G. We call any function f of the form f : σ ∪ { ∅ } → { formations of groups } , where f (∅) ≠ ∅ , a generalized formation σ-function and we put B L F σ (f) = (G | G / R σ (G) ∈ f (∅) and G / F { g σ i } (G) ∈ f (σ i) for all σ i ∈ σ + (G)). If for some generalized formation σ-function f we have F = B L F σ (f) , then we say that the class F is Baer-σ-local and f is a generalized σ-local definitionof F. In this article, we describe basic properties, examples, and some applications of Baer-σ-local formations. In particular, we prove that the set of all Baer-σ-local formations containing all nilpotent groups forms a subsemigroup of the semigroup of all formations G G (Theorem 1.13) and the set of all Baer-σ-local formations containing all σ-nilpotent groups is a right ideal in G G (Theorem 1.16). [ABSTRACT FROM AUTHOR]

Let F denote a class of groups. A maximal subgroup M of G is called F -abnormal provided G/MG ∉ F . We say that (K, H) is an F -abnormal pair of G provided K is a maximal F -abnormal subgroup of H. Let Σ = {G0 ≤ G1 ≤ G2 ≤ ... ≤ Gn} be a subgroup series of G. A subgroup H of G is said to be Σ- F -embedded in G if H either covers or avoids every F -abnormal pair (K, H) such that Gi−1≤ K < H ≤ Gi for some i ∈ {0, 1, ..., n}. In this paper, some new characterizations of p-supersoluble and p-soluble are given by discussing the properties of Σ- F -embedded of subgroups. [ABSTRACT FROM AUTHOR]

A nonempty class X of finite groups is called complete if it is closed under taking subgroups, homomorphic images and extensions. We consider two definitions of submaximal X -subgroups suggested by H. Wielandt and discuss which one better suits the task of determining maximal X -subgroups. We prove that these definitions are not equivalent yet the Wielandt–Hartley theorem holds true for either definition of X -submaximality. We also give some applications of the strong version of the Wielandt–Hartley theorem. [ABSTRACT FROM AUTHOR]

Let σ = { σ i : i ∈ I } be a partition of the set P of all prime numbers. A subgroup A of a finite group G is called σ-subnormal in G if there is a chain of subgroups A = A 0 ⊆ A 1 ⊆ ⋯ ⊆ A n = G with A i − 1 normal in Ai or A i / Core A i (A i − 1) is a σ j -group for some j ∈ I , for every 1 ≤ i ≤ n. In this paper, the description of the unique smallest σ-subnormal subgroup of a σ-soluble group containing a given subgroup is obtained. [ABSTRACT FROM AUTHOR]

A subgroup H of a finite group G is weakly supplemented in G if there exists a proper subgroup K of G such that G = H K . In the paper, we present some sufficient and necessary conditions for a finite group to be p-nilpotent and solvable by using some weakly supplemented minimal subgroups. As applications, we extend some known results. [ABSTRACT FROM AUTHOR]

Let G be a finite group, p a prime, P a Sylow p-subgroup of G and d a power of p such that 1 < d < | P |. Let G p * denote the unique smallest normal subgroup of G for which the corresponding factor group is abelian of exponent dividing p – 1. Let F 1 , F 2 , F 3 be classes of all p-groups, p-nilpotent groups and p-supersolvable groups, respectively, G F be the F -residual of G. Let E ∈ { G F 1 , (G p *) F 1 , G F 2 , G F 3 }. We prove that if H ∩ E is weakly s-semipermutable in G for all normal subgroups H of P with | H | = d , then either G ∈ F 3 and G ∈ F 2 if p is the least prime factor of | G | , or else | P ∩ E | > d ; and for all subgroups H of P with | H | = d and 4 (if p = 2), then G ∈ F 3 and G ∈ F 2 if p is the least prime factor of | G |. Many known theorems become Corollaries of our results. Communicated by Peter Sin [ABSTRACT FROM AUTHOR]

It is known that there exists a first-order sentence that holds in a finite group if and only if the group is soluble. Here it is shown that the corresponding statements with 'solubility' replaced by 'nilpotence' and 'perfectness', among others, are false. These facts present difficulties for the study of pseudofinite groups. However, a very weak form of Frattini's theorem on the nilpotence of the Frattini subgroup of a finite group is proved for pseudofinite groups. [ABSTRACT FROM AUTHOR]

Let n be a positive integer, and let σ = { σ i ∣ i ∈ I } be a partition of the set of all primes. It is shown that every law of the lattice of all formations is fulfilled in the lattice of all n-multiply σ -local formations of finite groups. This result implies that the lattice of all n-multiply σ -local formations of finite groups is modular but not distributive for any nonnegative integer n. Let F and H be n-multiply σ -local formations of finite groups such that H ⊆ F . Denote by F / n σ H the lattice of all n-multiply σ -local formations M such that H ⊆ M ⊆ F . If M ⊂ F and the lattice F / n σ M consists of only two elements then M is called a maximal n-multiply σ -local subformation of F . The properties of the intersection of the maximal n-multiply σ -local subformations of F are described. [ABSTRACT FROM AUTHOR]

We finish the classification, begun in two earlier papers, of all simple fusion systems over finite nonabelian p-groups with an abelian subgroup of index p. In particular, this gives many new examples illustrating the enormous variety of exotic examples that can arise. In addition, we classify all simple fusion systems over infinite nonabelian discrete p-toral groups with an abelian subgroup of index p. In all of these cases (finite or infinite), we reduce the problem to one of listing all pG-modules (for G finite) satisfying certain conditions: a problem which was solved in the earlier paper [15] using the classification of finite simple groups. [ABSTRACT FROM AUTHOR]

The finite group G is called a CLT-group if there exists a subgroup H of G of order d, for every divisor d of the order of G. It is said to be a generalized CLT-group if, for every divisor d of |G|, there exists a subgroup H of G of order d, such that H satisfies some normality condition. In this paper, generalized CLT-groups are characterized. [ABSTRACT FROM AUTHOR]

Let $\mathfrak{F}$ be a class of finite groups and $G$ a finite group. Let ${\mathcal{L}}_{\mathfrak{F}}(G)$ be the set of all subgroups $A$ of $G$ with $A^{G}/A_{G}\in \mathfrak{F}$. A chief factor $H/K$ of $G$ is $\mathfrak{F}$ - central in $G$ if $(H/K)\rtimes (G/C_{G}(H/K))\in \mathfrak{F}$. We study the structure of $G$ under the hypothesis that every chief factor of $G$ between $A_{G}$ and $A^{G}$ is $\mathfrak{F}$ -central in $G$ for every subgroup $A\in {\mathcal{L}}_{\mathfrak{F}}(G)$. As an application, we prove that a finite soluble group $G$ is a PST -group if and only if $A^{G}/A_{G}\leq Z_{\infty }(G/A_{G})$ for every subgroup $A\in {\mathcal{L}}_{\mathfrak{N}}(G)$ , where $\mathfrak{N}$ is the class of all nilpotent groups. [ABSTRACT FROM AUTHOR]

In this article, we give a complete classification of finite groups whose second maximal subgroups are all abelian. [ABSTRACT FROM AUTHOR]

A subgroup S of a group G is called a p-sylowizer of a p-subgroup R in G if S is maximal in G with respect to having R as its Sylow p-subgroup. The main aim of this paper is to investigate the influence of p-sylowizers on the structure of finite groups. We obtained some new characterizations of p-nilpotent and supersolvable groups by the permutability of the p-sylowizers of some p-subgroups. In addition, we determined all p-sylowizers of arbitrary p-subgroups for the supersolvable groups. [ABSTRACT FROM AUTHOR]

Let G be a finite group and let π be a set of primes. We prove that G has a solvable π-Hall subgroup if and only if G has a {p, q}-Hall subgroup for every pair p, q ∈ π. [ABSTRACT FROM AUTHOR]

Suppose that G is a finite group and H is a subgroup of G. Subgroup H is said to be weakly ℳ -supplemented in G if there exists a subgroup B of G such that (1) G = HB, and (2) if H1/HG is a maximal subgroup of H/HG, then H1B = BH1 < G, where HG is the largest normal subgroup of G contained in H. We fix in every noncyclic Sylow subgroup P of G a subgroup D satisfying 1 < ∣D∣ < ∣P∣ and study the p-nilpotency of G under the assumption that every subgroup H of P with ∣H∣ = ∣D∣ is weakly ℳ -supplemented in G. Some recent results are generalized. [ABSTRACT FROM AUTHOR]

In this paper, we give a complete classification of the finite groups G whose second maximal subgroups are cyclic

Suppose that the finite group G = A B is a mutually permutable product of two p-soluble subgroups A and B. By using the Wielandt lenghts of Sylow p-subgroups of A and B, we present a new bound of the p-length of G. Some known results are improved. [ABSTRACT FROM AUTHOR]

A subgroup A of a finite group G is said to be NS-supplemented in G, if there exists a subgroup B of G such that G=AB and whenever X is a normal subgroup of A and p ∈ π (B) , there exists a Sylow p-subgroup Bp of B such that X B p = B p X . In this paper, we prove the supersolubility of a group G in the following cases: every non-cyclic Sylow subgroup of G is NS-supplemented in G; G is soluble and all maximal subgroups of every non-cylic Sylow subgroup of G are NS-supplemented in G. The solubility of a group with NS-supplemented maximal subgroups is obtained. [ABSTRACT FROM AUTHOR]

We construct examples that show the difference in behavior of π-maximal and π-submaximal subgroups under group homomorphisms. [ABSTRACT FROM AUTHOR]

A subgroup H of a finite group G is said to be Hall subnormally embedded in G if there is a subnormal subgroup N of G such that H is a Hall subgroup of N. A Schmidt group is a finite non-nilpotent group whose all proper subgroups are nilpotent. We prove the nilpotency of the second derived subgroup of a finite group in which each Schmidt subgroup is Hall subnormally embedded. [ABSTRACT FROM AUTHOR]

We provide structural criteria for some finite factorised groups G = A B when the conjugacy class sizes in G of certain π -elements in A ∪ B are either π -numbers or π ′ -numbers, for a set of primes π . In particular, we extend for products of groups some earlier results. [ABSTRACT FROM AUTHOR]

In this paper, we study theorems related to Sylow intersections and we strengthen a result of Robinson (Proc Am Math Soc 90(1):21–24,1984) and also a result of Brewster and Hauck (J Algebra 206:261–292, 1998). [ABSTRACT FROM AUTHOR]

In this paper, we introduce the concept of sse-embedded subgroups of finite groups and present some new characterizations of solubility of finite groups using the sse-embedding property of subgroups. Furthermore, we discuss the sse-embedded subgroups in finite nonabelian simple groups. Some previously known results are generalized and unified. [ABSTRACT FROM AUTHOR]

Assume that is a class of finite groups. A normal subgroup E is Φ- hypercentral in G if E ≤ Z Φ (G), where Z Φ (G) denotes the Φ-hypercentre of G. We call a subgroup H is p-embedded in G, if there exists a p-nilpotent subgroup B of G such that Hp ∈ Sylp(B) and B is p-supplemented in G, where Hp is a Sylow p-subgroup of H. In this paper, the main result is that: Let E be a normal subgroup of G. For all p ∈ π(F*(E)) and every noncyclic Sylow p-subgroup P of F*(E), if there is a prime power pα such that 1 < pα ≤ | P | and every subgroup H of P with | H| = pα is p-embedded in G, then E is Φ-hypercentral in G. [ABSTRACT FROM AUTHOR]

Let B be a p-block of the finite group G. We observe that the p-fusion of G constrains the module structure of B: Any basis of B that is closed under the left and right multiplications of a chosen Sylow p-subgroup S of G must in fact form a semicharacteristic biset for the fusion system on S induced by G. The parameterization of such semicharacteristic bisets can then be applied to relate the module structure and defect theory of B. [ABSTRACT FROM AUTHOR]

Suppose that G is a finite group and H is a subgroup of G. H is said to be weakly s-semipermutable in G if there are a subnormal subgroup T of G and an s-semipermutable subgroup H ssG of G contained in H such that G = H T and H ∩ T ≤ H ssG ; H is said to be an ss-quasinormal subgroup of G if there is a subgroup B of G such that G = H B and H permutes with every Sylow subgroup of B. We fix in every non-cyclic Sylow subgroup P of G some subgroup D satisfying 1 < | D | < | P | and study the structure of G under the assumption that every subgroup H of P with | H | = | D | is either weakly s-semipermutable or ss-quasinormal in G. Some recent results are generalized and unified. [ABSTRACT FROM AUTHOR]

In this paper, we study the relationships between the elementary abelian regular subgroups and the Sylow 2-subgroups of their normalisers in the symmetric group Sym ( F 2 n) , in view of the interest that they have recently raised for their applications in symmetric cryptography. [ABSTRACT FROM AUTHOR]

Let σ be some partition of the set of all primes and a complete Hall σ-set of a finite group G. A subgroup H of G is said to be σ-conditionally permutable in G if for any subgroup there exists an element such that In this article, we investigate the influence of σ-conditionally permutable subgroups on the structure of finite groups. [ABSTRACT FROM AUTHOR]

Let G be a finite group and let p be prime dividing. In this article, we supply some sufficient conditions for G to be p-nilpotent (see Theorem 1.2) as an extension of the main theorem of Li et al. (J. Group Theor. 20(1): 185–192, 2017). [ABSTRACT FROM AUTHOR]

A subgroup H of a finite group G is said to be nearly s-embedded in G if G has an s-permutable subgroup T and an s-semipermutable subgroup HssG contained in H such that and , where HsG is the intersection of all s-permutable subgroups of G containing H. In this paper, we present an extension of a result on p-supersolvability of finite groups under the assumption that maximal subgroups of some Sylow p-subgroups are nearly s-embedded. As applications, we give several new criteria for a finite group to be p-nilpotent and supersolvable. [ABSTRACT FROM AUTHOR]

In this paper we study the saturated fusion systems over a direct product of the extraspecial group of order p3 of exponent p and a finite abelian p-group. The result provides some new exotic fusion systems, whose unique components are isomorphic to the exotic fusion systems over 7 + 1 + 2 found by Ruiz and Viruel. [ABSTRACT FROM AUTHOR]

Let G be a finite group and δ (G) denote the number of conjugate classes of allnon-cyclic subgroups of G. The symbol π (G) denotes the set of the prime divisors of |G|. In Meng and Li (Sci Sin Math 44:939–944, 2014), it was proved that for a finite non-cyclic solvable group G, one always has δ (G) ≥ 2 | π (G) | - 2 . The groups with δ (G) ≤ | π (G) | + 1 always are solvable and have been complete classified. Moreover, it was showed that a finite non-solvable group G with δ (G) = | π (G) | + 2 is isomorphic to A 5 or SL(2, 5). In this paper, we investigate the finite solvable groups with δ (G) = | π (G) | + 2 . For convenience, a group G is said to be a δ π 2 -group if δ (G) = | π (G) | + 2 . In particular, we give a completely classification of the δ π 2 -groups with | π (G) | = 3 , 4 . [ABSTRACT FROM AUTHOR]

In this paper, some equivalent conditions on a finite group G will be given in order the group G to have a unique, and necessarily cyclic, subgroup of order m, where m is a fixed divisor of |G|. These conditions are about "the number of cyclic subgroups of particular orders" and "the number of elements of particular orders". [ABSTRACT FROM AUTHOR]

Let σ = {σi | i ∈ I} be some partition of the set of all primes P. A set H of subgroups of G is said to be a complete Hallσ-set of G if every member ≠ 1 of H is a Hall σi-subgroup of G, for some i ∈ I, and H contains exactly one Hall σi-subgroup of G for every σi ∈ σ(G). A subgroup H of G is said to be: σ-permutable or σ-quasinormal in G if G possesses a complete Hall σ-set H such that HAx = AxH for all A ∈ H and x ∈ G: σ-subnormal in G if there is a subgroup chain A = A0 ≤ A1 ≤ · · · ≤ At = G such that either A i − 1 ⊲ _ A i or Ai=(Ai-1)Ai is a finite σi-group for some σi ∈ σ for all i = 1;:::; t. If Mn < Mn-1 < · · · < M1 < M0 = G, where Mi is a maximal subgroup of Mi-1, i = 1; 2;...; n, then Mn is said to be an n-maximal subgroup of G. If each n-maximal subgroup of G is σ-subnormal (σ-quasinormal, respectively) in G but, in the case n > 1, some (n−1)-maximal subgroup is not σ-subnormal (not σ-quasinormal, respectively) in G, we write mσ(G) = n (mσq(G) = n, respectively). In this paper, we show that the parameters mσ(G) and mσq(G) make possible to bound the σ-nilpotent length lσ(G) (see below the definitions of the terms employed), the rank r(G) and the number |П(G)| of all distinct primes dividing the order |G| of a finite soluble group G. We also give the conditions under which a finite group is σ-soluble or σ-nilpotent, and describe the structure of a finite soluble group G in the case when mσ(G) = |П(G)|. Some known results are generalized. [ABSTRACT FROM AUTHOR]

Throughout this paper, all groups are finite. σ = { σ i ∣ i ∈ I } is some partition of the set of all primes P , and σ (n) = { σ i ∣ σ i ∩ π (n) ≠ ∅ } for any n ∈ N . The natural numbers n and m are called σ -coprime if σ (n) ∩ σ (m) = ∅ . Let t > 1 be a natural number and let F be a class of groups. Then we say that F is Γ t σ -closed (respectively weakly Γ t σ -closed) provided F contains each finite group G which satisfies the following conditions: G has subgroups A 1 , ... , A t ∈ F such that G = AiAj for all i ≠ j ; The indices | G : N G (A 1) | , ... , | G : N G (A t) | (respectively the indices | G : A 1 | , ... , | G : A t - 1 | , | G : N G (A t) | ) are pairwise σ -coprime. We study properties and some applications of (weakly) Γ t σ -closed classes of finite groups. [ABSTRACT FROM AUTHOR]

A subgroup H of a finite group G is weakly-supplemented in G if there exists a proper subgroup K of G such that G = HK. In this paper, some interesting results with weakly-supplemented minimal subgroups or Sylow subgroups of G are obtained. [ABSTRACT FROM AUTHOR]

Let G be a finite group, σ = { σ i | i ∈ I } be a partition of the set of all primes P and σ (G) = { σ i | σ i ∩ π (G) ≠ ∅ } . G is said to be σ -primary if | σ (G) | ≤ 1 ; σ -soluble if every chief factor of G is σ -primary. A set H of subgroups of G is said to be a complete Hall σ -set of G if every non-identity member of H is a Hall σ i -subgroup of G for some σ i and H contains exactly one Hall σ i -subgroup for every σ i ∈ σ (G) . G is said to be σ -full if G possesses a complete Hall σ -set; σ -nilpotent if G has a complete Hall σ -set H = { H 1 , H 2 , ... , H t } such that G = H 1 × H 2 × ⋯ × H t ; N σ -critical (resp. N -critical) if G is not σ -nilpotent (resp. nilpotent) but all proper subgroups of G are σ -nilpotent (resp. nilpotent). An N -critical group is also called a Schmidt group. In this paper, we first prove that every N σ -critical group is σ -soluble. This result gives a positive answer to a recent open problem of Skiba. We also prove that N σ -critical groups are also Schmidt groups and so the structure of N σ -critical groups is obtained. [ABSTRACT FROM AUTHOR]

A group G has finite (or Prüfer or special) rank if every finitely generated subgroup of G can be generated by r elements and r is the least integer with this property. The aim of this paper is to prove the following result: assume that G = A B is a group which is the mutually permutable product of the abelian subgroups A and B of Prüfer ranks r and s, respectively. If G is locally finite, then the Prüfer rank of G is at most r + s + 3 . If G is an arbitrary group, then the Prüfer rank of G is at most r + s + 4 . [ABSTRACT FROM AUTHOR]

Let G be a finite group, let π be a set of primes and let p be a prime. We characterize the existence of a normal Hall π‐subgroup in G in terms of the order of products of certain elements of G. This theorem generalizes a characterization of A. Moretó and the second author by using the orders of products of elements for those groups having a normal Sylow p‐subgroup. As a consequence, we also give a π‐decomposability criterion for a finite group also by means of the orders of products. [ABSTRACT FROM AUTHOR]

Suppose that G and A are finite groups such that A acts coprimely on G via automorphisms. It is interesting to investigate the structure and properties of G when we impose some restrictions on its maximal A-invariant subgroups. More precisely, we prove the solvability of G when certain maximal A-invariant subgroups are nilpotent, when all maximal A-invariant subgroups are supersolvable, or when certain arithmetic conditions are imposed on non-nilpotent maximal A-invariant subgroups. [ABSTRACT FROM AUTHOR]

In this paper, we give some new conditions under which a normal subgroup E of a finite group G is hypercyclically embedded in G, that is, every chief factor of G below E is cyclic. [ABSTRACT FROM AUTHOR]

Throughout this paper, all groups are finite. Let be some partition of the set of all primes. If n is an integer, the symbol denotes the set ; and. We call any function f of the form a formation σ-function, and we put If for some formation σ-function f we have , then we say that the class is σ-local and f is a σ-local definition of. We suppose that every formation is 0-multiply σ-local; for n > 0, we say that the formation is n-multiply σ-local provided either is the class of all identity groups or where is multiply σ-local for all. In this paper, we describe some properties and examples of n-multiply σ-local formations. In particular, we prove that the Gaschütz product of any two n-multiply σ-local formations is also n-multiply σ-local. We also consider one application of such formations in the theory of finite factorizable groups. [ABSTRACT FROM AUTHOR]

Let be a partition of the set of all primes and G a finite group. A set of subgroups of G is said to be a complete Hall σ-set of G if every member of is a Hall σi-subgroup of G for some and contains exactly one Hall σi-subgroup of G for every i such that. A group is said to be σ-primary if it is a finite σi-group for some i. A subgroup A of G is said to be: σ-permutable in G if G possesses a complete Hall σ-set such that for all and all ; -subnormal in G if there is a subgroup chain such that either or is σ-primary for all. We say that a subgroup H of G is: m-σ-permutable in G if for some modular subgroup A and σ-permutable subgroup B of G; weakly m-σ-permutable in G if there are an m-σ-permutable subgroup S and a σ-subnormal subgroup T of G such that G = HT and. We study G assuming that some subgroups of G are weakly m-σ-permutable in G. [ABSTRACT FROM AUTHOR]

A subgroup H of a finite group G is weakly-supplemented in G if there exists a proper subgroup K of G such that G = HK. In this paper, some interesting results with weakly-supplemented minimal subgroups to a smaller subgroup of G are obtained. [ABSTRACT FROM AUTHOR]