Nilpotent Hall Subgroups Research Articles

Variations on results on orders of products in finite groups

In 2014, Baumslag and Wiegold proved that a finite group G is nilpotent if and only if o(xy) = o(x)o(y) for every x, y ∈ G with (o(x), o(y)) = 1. This has led to a number of results that characterize the nilpotence of a group (or the existence of nilpotent Hall subgroups, or the existence of normal Hall subgroups) in terms of prime divisors of element orders. Here, we look at these results with a new twist. The first of our main results asserts that G is nilpotent if and only if o(xy) ⩽ o(x)o(y) for every x, y ∈ G of prime power order with (o(x), o(y)) = 1. As an immediate consequence, we recover the Baumslag–Wiegold theorem. The proof of this result is elementary. We prove some variations of this result that depend on the classification of finite simple groups.

On normal subgroups of M-groups

The normal embedding problem of finite solvable groups into [Formula: see text]-groups was studied. It was proved that for a finite solvable group [Formula: see text], if [Formula: see text] has a special normal nilpotent Hall subgroup, then [Formula: see text] cannot be a normal subgroup of any [Formula: see text]-group; on the other hand, if [Formula: see text] has a maximal normal subgroup which is an [Formula: see text]-group, then [Formula: see text] can occur as a normal subgroup of an [Formula: see text]-group under some suitable conditions. The results generalize the normal embedding theorem on solvable minimal non-[Formula: see text]-groups to arbitrary [Formula: see text]-groups due to van der Waall, and also cover the famous counterexample given by Dade and van der Waall independently to the Dornhoff’s conjecture which states that normal subgroups of arbitrary [Formula: see text]-groups must be [Formula: see text]-groups.

Finite groups with permutable complete Wielandt sets of subgroups

Abstract Let 𝒲 be a set of nilpotent Hall subgroups of a finite group G. We say that 𝒲 is a Wielandt set if for each prime p dividing |G| there is exactly one subgroup W ∈ 𝒲 such that p divides |W|. In this paper, we study the influence of Wielandt sets on the structure of G. In particular, we give conditions under which a normal subgroup of a finite group is hypercyclically embedded.

ON SYLOW NORMALIZERS OF FINITE GROUPS

The paper considers the influence of Sylow normalizers, i.e. normalizers of Sylow subgroups, on the structure of finite groups. In the universe of finite soluble groups it is known that classes of groups with nilpotent Hall subgroups for given sets of primes are exactly the subgroup-closed saturated formations satisfying the following property: a group belongs to the class if and only if its Sylow normalizers do so. The paper analyzes the extension of this research to the universe of all finite groups.

Sylow numbers and nilpotent Hall subgroups

Let π be a set of primes and G a finite group. We characterize the existence of a nilpotent Hall π-subgroup of G in terms of the number of Sylow subgroups for the primes in π.

Groups with two Sylow numbers are the product of two nilpotent Hall subgroups

We have recently found a characterization of the existence of nilpotent Hall subgroups in a finite group. Using this result, we prove the statement in the title. This is a strong form of a conjecture of J. Zhang.

Generalizations of Groups in which Normality Is Transitive

A group G is called a Hall𝒳-group if G possesses a nilpotent normal subgroup N such that G/N′ is an 𝒳-group. A group G is called an 𝒳o-group if G/Φ(G) is an 𝒳-group. The aim of this article is to study finite solvable Hall𝒳-groups and 𝒳o-groups for the classes of groups 𝒯, 𝒫𝒯, and 𝒫𝒮𝒯. Here 𝒯, 𝒫𝒯, and 𝒫𝒮𝒯 denote, respectively, the classes of groups in which normality, permutability, and Sylow-permutability are transitive relations. Finite solvable 𝒯-groups, 𝒫𝒯-groups, and 𝒫𝒮𝒯-groups were globally characterized, respectively, in Gaschütz (1957), Zacher (1964), and Agrawal (1975). Here we arrive at similar characterizations for finite solvable Hall𝒳-groups and 𝒳o-groups where 𝒳 ∈ {𝒯, 𝒫𝒯, 𝒫𝒮𝒯}. A key result aiding in the characterization of these groups is their possession of a nilpotent residual which is a nilpotent Hall subgroup of odd order. The main result arrived at is Hall𝒫𝒮𝒯 = 𝒯o for finite solvable groups.

SATURATED FORMATIONS CLOSED UNDER SYLOW NORMALIZERS

In this article we show that a finite soluble group possesses nilpotent Hall subgroups for well-defined sets of primes if and only if its Sylow normalizers satisfy the same property. In fact, this property of groups provides a characterization of the subgroup-closed saturated formations, whose elements are characterized by the Sylow normalizers belonging to the class, in the universe of all finite soluble groups.

AUTOMORPHISMS, FACTORIZATIONS AND SYLOW-TYPE THEOREMS

In this paper finite groups with restrictions on certain subgroups are studied. A generalization of Wielandt's theorem on groups having a nilpotent Hall subgroup is obtained. A conjecture by S. A. Chunikhin on the nonsimplicity of a finite group having two classes of nontrivial elements of odd order with relatively prime centralizer indexes is also confirmed.Bibliography: 22 titles.

Finite groups admitting the extensions of the automorphisms of a maximal elementary abelian 2-subgroup

1. INTRODUCTION AND STATEMENT OF THE THEOREM In [lo], Higman described the structure of a finite p-group admitting an automorphism which cyclically permutes the subgroups of order p. Such a group is either a direct sum of cyclic groups of the same order, a generalized quaternion group, or a Suzuki 2-group (defined in [lo]). Several authors have considered the structure of a finite p-group whose automorphism group acts transitively on the elements of order p of the group. (See [S] and the references cited in it.) Gaschiitz and Yen [S] have shown that if G is a group of odd order and, if for each prime divisor p of the order of G, the automorphism group of G acts transitively on the elements of G of order p, then G is a cyclic extension of a nilpotent Hall subgroup. Here, we consider a variation of the situations mentioned above. We say that a finite group G has the (extension) property B(p) for the prime p if the automorphisms of some elementary abelian p-subgroup E of G of the largest possible order can be obtained as the restrictions to E of the automorphisms of G. Our object here is to prove the following: THEOREM. Let G be a finite group possessing the property g(2), and let (G#)S E W,(G). (A) If O,(G) # (1) and O(G) = (1 ), then one of

Intersections of nilpotent Hall subgroups

Intersections of nilpotent Hall subgroups

Finite groups admitting a fixed-point-free automorphism

In 1959 J. Thompson proved that a finite group admitting a fixed-point-free automorphism 0 of prime order n must be nilpotent. The aim of the present paper is to generalize this fine result by weakening the assumption on n being a prime into proper requirements for the fixed points of the nontrivial powers of rr (clearly, if 12 is a prime, all these automorphisms are fixed-point-free). The starting point (Lemma I) is a generalization of a well-known result (see [4], IX, 4. e) on the fixed points of some groups of automorphisms operating on Abelian groups. We apply this lemma to prove the following statement (Theorem 1) : Let G be a jinite group admitting (r us a f.p.f. automorphism. Let II be a nilpotent Hall subgroup of G, H having Abelian Sylow 2-subgroups. Assume that the jixed points of the nontriviul powers of o are all in H. Then H has a nilpotent normal complement in G. We derive next a somewhat symmetrical form of the preceding result, applying to solvable groups (Theorem 2). Then we make some restrictions on n. : the following theorem (Theorem 3) can also be interpreted as a generalization of Thompson’s result: Let G be afinitegroup admitting (I as a f.p.f. automorphism of order n, n being a product of dt@rent primes. Assume that the fixed points of the nontrivial powers of (T are all in the same nilpotent Hall subgroup of G. Then G is nilpotent. Other statements (Theorem 4 and Corollary 2) are improvements of our previous results for n = pq, a product of two different primes, [5]. In the last section we point out how Lemma 1 (possibly in a slightly different form) can also be applied to make some well-known proofs of the literature remarkably simpler.