The nilpotence class of the Frattini subgroup

Ap-group of sufficiently large nilpotence class cannot occur as a normal subgroup contained in the Frattini subgroup of any finite group. The Frattini subgroup of a group of order Π _pi ^αi with max α _i at least 3, has nilpotence class at most 1/2 (max α _i − 1). The Frattini subgroup of at-group is abelian. The occurrence of groups of orderp ⁴ as normal subgroups contained in Frattini subgroups is investigated. National Science Foundation Science Faculty Fellow, University of Cincinnati     

Two-generator Frattini subgroups of finitep-groups

This paper deals with nonabelianp-groupsT (p a prime andp>2) which are either metacyclic or Redei. These groups are classified into those which are Frattini subgroups of a finitep-groupG and those which are not. Finally, it is shown that a nonabelian two-generator group of orderp ⁿ (n>4) which is the Frattini subgroup of ap-group must be metacyclic. This work is contained in the author’s dissertation.     

Frattini subgroups ofp-closed andp-supersolvable groups

A nonabelianp-group with cyclic center cannot occur as a normal subgroup contained in the Frattini subgroup of ap-closed group. If a nonabelian normal subgroup of orderp ⁿ and nilpotence classk is contained in the Frattini subgroup of ap-closed group, then its exponent is a divisor ofp ^n−k . This fact is used to derive a relation among the order, number of generators, exponent, and class of the Frattini subgroup, forp-groups. Finally, it is conjectured that a nonabelianp-group having center of orderp cannot occur as a normal subgroup contained in the Frattini subgroup of any finite group. A proof is given forp-supersolvable groups.     

Finite 2-groups with a nonabelian Frattini subgroup of order 16

According to a classical result of Burnside, if G is a finite 2-group, then the Frattini subgroup Φ(G) of G cannot be a nonabelian group of order 8. Here we study the next possible case, where G is a finite 2-group and Φ(G) is nonabelian of order 16. We show that in that case Φ(G) ≅ M × C₂, where M ≅ D₈ or M ≅ Q₈ and we shall classify all such groups G (Theorem A). Received: 16 February 2005; revised: 7 March 2005     

On a class of groups of prime-power order

LetG be a finitep-group,d(G)=dimH ¹ (G, Z _p) andr(G)=dimH ²(G, Z_p). Thend(G) is the minimal number of generators ofG, and we say thatG is a member of a classG _p of finitep-groups ifG has a presentation withd(G) generators andr(G) relations. We show that ifG is any finitep-group, thenG is the direct factor of a member ofG _p by a member ofG _p .     

Applications of dimension subgroups to the power structure ofp-groups

Dimension subgroups in characteristicp are employed in the study of the power structure of finitep-groups. We show, e.g., that ifG is ap-group of classc (p odd) andk=⌜log _p ((c+1)/(p−1))⌝, then, for alli, any product ofp ^i+k th powers inG is ap ⁱ th power. This sharpens a previous result of A. Mann. Examples are constructed in order to show that our constantk is quite often the best possible, and in any case cannot be reduced by more than 1. Partially supported by MPI funds. This author is a member of GNSAGA-CNR. Partially supported by a Rothschild Fellowship.     

Epimorphisms of demushkin groups

A necessary condition that a continuous epimorphism from a Demushkin groupG onto a finitep-groupH can be factored epimorphically through a free prop-groupS-is given, which is sufficient whenH is abelian of exponentp ^m≠2,m depending onG, 1≦m≦∞. In particular a free prop-factor groupS toG can have rank at most one half rankG. Application is made to embedding problems over localp-adic fields.     

On the Schur multipliers of finite p-groups of given coclass

In this paper we obtain bounds for the order and exponent of the Schur multiplier of a p-group of given coclass. These are further improved for p-groups of maximal class. In particular, we prove that if G is p-group of maximal class, then |H ₂(G, ℤ)| < |G| and expH ₂(G, ℤ) ≤ expG. The bound for the order can be improved asymptotically.     

Normal <Emphasis Type="Italic">π</Emphasis>-complements for finite groups

Let G be a finite group. For a finite p-group P the subgroup generated by all elements of order p is denoted by Ω₁(p). Zhang [5] proved that if P is a Sylow p-subgroup of G, Ω₁(P) ≦ Z(P) and N _G (Z(P)) has a normal p-complement, then G has a normal p-complement. The object of this paper is to generalize this result. This paper was partly supported by Hungarian National Foundation for Scientific Research Grant # T049841 and T038059.     

The fitting length of solvableH pn-groups

For a (finite) groupG and some prime powerp ⁿ, theH _p ⁿ -subgroupH _pn (G) is defined byH _p ⁿ (G)=〈xεG|x ^pn≠1〉. A groupH≠1 is called aH _p ⁿ -group, if there is a finite groupG such thatH is isomorphic toH _p ⁿ (G) andH _p ⁿ (G)≠G. It is known that the Fitting length of a solvableH _p ⁿ -group cannot be arbitrarily large: Hartley and Rae proved in 1973 that it is bounded by some quadratic function ofn. In the following paper, we show that it is even bounded by some linear function ofn. In view of known examples of solvableH _p ⁿ -groups having Fitting lengthn, this result is “almost” best possible.     

Omega subgroups of powerful <Emphasis Type="Italic">p</Emphasis>-groups

Let G be a powerful finite p-group. In this note, we give a short elementary proof of the following facts for all i ≥ 0: (i) exp Ω_i(G) ≤ p ⁱ for odd p, and expΩ_i(G) ≤ 2 ⁱ⁺¹ for p = 2; (ii) the index |G: G ^{p i}| coincides with the number of elements of G of order at most p ⁱ. Supported by the Spanish Ministry of Science and Education, grant MTM2004-04665, partly with FEDER funds, and by the University of the Basque Country, grant UPV05/99.     

On division rings generated by polycyclic groups

LetD=F(G) be a division ring generated as a division ring by its central subfieldF and the polycyclic-by-finite subgroupG of its multiplicative group, letn be a positive integer and letX be a finitely generated subgroup of GL(n, D). It is implicit in recent works of A. I. Lichtman thatX is residually finite. In fact, much more is true. If charD=p≠0, then there is a normal subgroup ofX of finite index that is residually a finitep-group. If charD=0, then there exists a cofinite set π=π(X) of rational primes such that for eachp in π there is a normal subgroup ofX of finite index that is residually a finitep-group.     

S-quasinormallity of finite groups

Let d be the smallest generator number of a finite p-group P, and let ℳ _d (P) = {P ₁,..., P _d } be a set of maximal subgroups of P such that ∩ _i=1 ^d P _i =Φ(P). In this paper, the structure of a finite group G under some assumptions on the S-quasinormally embedded or SS-quasinormal subgroups in ℳ _d (P), for each prime p, and Sylow p-subgroups P of G is studied.     

The nilpotency class of the unit group of a modular group algebra II

LetG be a finitep-group, and letU(G) be the group of units of the group algebraFG, whereF is a field of characteristicp. It is shown that, if the commutative subgroup ofG has order at leastp ², then the nilpotency class ofU(G) is at least 2p−1. The authors are grateful to the Dipartimento di Matematica of the Universita di Trento, and to the Mathematical Institute of the University of Oxford, for their hospitality while this paper was being written. Then are also grateful to Robert Sandling, for communication of results, and problems, prior to publication.     

Nonnormal and minimal nonabelian subgroups of a finite group

Let G be a finite p-group. If p = 2, then a nonabelian group G = Ω₁(G) is generated by dihedral subgroups of order 8. If p > 2 and a nonabelian group G = Ω₁(G) has no subgroup isomorphic to S_p²{\Sigma _{{p^2}}}, a Sylow p-subgroup of the symmetric group of degree p ², then it is generated by nonabelian subgroups of order p ³ and exponent p. If p > 2 and the irregular p-group G has < p nonabelian subgroups of order p ^p and exponent p, then G is of maximal class and order p ^p+1. We also study in some detail the p-groups, containing exactly p nonabelian subgroups of order p ^p and exponent p. In conclusion, we prove three new counting theorems on the number of subgroups of maximal class of certain type in a p-group. In particular, we prove that if p > 2, and G is a p-group of order > p ^p+1, then the number of subgroups ≅ ΣS_p²{\Sigma _{{p^2}}} in G is a multiple of p.     

Finite groups in which permutability is a transitive relation on their frattini factor groups

Let G be a finite group. A PT-group is a group G whose subnormal subgroups are all permutable in G. A PST-group is a group G whose subnormal subgroups are all S-permutable in G. We say that G is a PT_o-group (respectively, a PST_o-group) if its Frattini quotient group G/Φ(G) is a PT-group (respectively, a PST-group). In this paper, we determine the structure of minimal non-PT_o-groups and minimal non-PST_o-groups.      

On Chernikovp-groups

We investigate extensions of divisible Abelianp-groups with minimality condition by means of a finitep-groupH and establish the conditions under which the problem of describing all nonisomorphic extensions of this sort is wild. All the nonisomorphic Chernikovp-groups are described whose factor-group with respect to the maximum divisible Abelian subgroup is a cyclic group of orderp ^s ,s≤2. Uzhgorod University, Uzhgorod. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 3, pp. 291–304, March, 1999.     

Vertices of simple modules and normal subgroups ofp-solvable groups

Let π be a set of prime numbers andG a finite π-separable group. Let θ be an irreducible π′-partial character of a normal subgroupN ofG and denote by I_π′ (G‖θ), the set of all irreducible π′-partial characters Φ ofG such that θ is a constituent of Φ_N. In this paper, we obtain some information about the vertices of the elements in I_π′ (G‖θ). As a consequence, we establish an analogue of Fong's theorem on defect groups of covering blocks, for the vertices of the simple modules (in characteristicsp) of a finitep-solvable group lying over a fixed simple module of a normal subgroup.     

Normal complements in modp-envelopes

SupposeG is a finitep-group andk is the field ofp elements, and letU be the augmentation ideal of the group algebrakG. We investigate whichp-groups,G, have normal complements in their modp-envelope,G ^*.G ^* is defined byG ^*={1−u∶u∈U}.     

Bounded pseudo-differential operators

This lecture gives an inside look into the proof of the continuity of pseudo-differential operators of orderm and typep, δ₁, δ₂ for 0≦p≦δ₁=1, 0≦p≦δ₂<1, andm/n≦p≦(δ₁+δ₂)/2. Applications are mentioned.