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.     

On the Frattini normal embeddability of products ofp-groups

ItH _i is a finite non-abelianp-group with center of orderp, for 1≦j≦R, then the direct product of theH _i does not occur as a normal subgroup contained in the Frattini subgroup of any finitep-group. If the Frattini subgroup Φ of a finitep-groupG is cyclic or elementary abelian of orderp ², then the centralizer of Φ inG properly contains Φ. Non-embeddability properties of products of groups of order 16 are established.     

On an embedding of certainp-groups

It is shown that ap-group with cyclic centre can be embedded in a finite group as a normal subgroup contained in its Frattini subgroup if and only if it is either an extraspecial 2-group of order at least 2⁷ or the central product of a cyclic groupQ of order ≧4 and an extraspecial groupE of order ≧2⁵, amalgamating Ω₁ (Q) and the centre ofE.     

On normalp-subgroups with large centers which cannot be contained in the Frattini subgroup

LetK be a characteristic subgroup of ap-groupH such thatH induces onK a sufficiently large group of automorphisms. ThenH cannot be embedded as a normal subgroup contained in the Frattini subgroup in any finite group. The groupH may have a large center without any characteristic subgroup ofH properly contained in it. Examples are given for suchH withZ(H) elementary abelian of arbitrary dimension.     

Frattini subgroups and supernilpotent groups

In the context of the problem of which nonabelianp-groups can occur as normal subgroups contained in Frattini subgroups, the family of supernilpotent groups (all maximal subgroups characteristic) is investigated. Results of this investigation are applied to the Frattini-embedding problem, incorporating recent work of A. R. Makan. The groups of order 2ⁿ (n ≦ 6) have been examined with respect to supernilpotence and their occurrence as normal subgroups contained in Frattini subgroups. Results of this examination are presented.     

Groups with few class sizes and the centraliser equality subgroup

LetG be ap-group whose conjugacy classes have at mostk sizes. We prove thatG is abelian-by-(exponentp ^k−1) (ifp=2, exponent 2 ^k−2). It follows that a 2-group with three class sizes is metabelian. Various other results on class sizes are proved, and some conjectures are formulated.     

On noninner automorphisms of 2-generator finite p-groups

A longstanding conjecture asserts that every finite nonabelian p-group admits a noninner automorphism of order p. In this paper we give some necessary conditions for a possible counterexample G to this conjecture, in the case when G is a 2-generator finite p-group. Then we show that every 2-generator finite p-group with abelian Frattini subgroup has a noninner automorphism of order p.     

Finite extensions of free pro-P groups of rank at most two

For a pro-p groupG, containing a free pro-p open normal subgroup of rank at most 2, a characterization as the fundamental group of a connected graph of cyclic groups of order at mostp, and an explicit list of all such groups with trivial center are given. It is shown that any automorphism of a free pro-p group of rank 2 of coprime finite order is induced by an automorphism of the Frattini factor groupF/F ^* . Finally, a complete list of automorphisms of finite order, up to conjugacy in Aut(F), is given. Supported by an NSERC grant. Supported by the Austrian Science Foundation.     

Finite groups in which normality is a transitive relation

Let G be a finite group. We say that G is a T₀-group, if its Frattini quotient group G/F(G)G/\Phi (G) is a T-group, where by a T-group we mean a group in which every subnormal subgroup is normal. We determine the structure of a non T₀-group G all of whose proper subgroups are T₀-groups.     

ON PRODUCTS OF GROUPS FOR WHICH NORMALITY IS A TRANSITIVE RELATION ON THEIR FRATTINI FACTOR GROUPS

Abstract

This paper is devoted to the study of groups with the property that the Frattini factor group is a T-group, i.e. a group in which every subnormal subgroup is normal. We give necessary and suffucient conditions for a direct product G = H x K of finite groups H and K to have such a property. Some structure theorems are also discussed.     

On the maximum number of fixed points in automorphisms of prime order of 2-(v,k,1) designs

In this paper, we study 2-(v, k, 1) designs with automorphisms of prime orderp, having the maximum possible number of fixed points. We prove an upper bound on the number of fixed points, and we study the structure of designs in which this bound is met with equality (such a design is called ap-MFP(v, k)). Several characterizations and asymptotic existence results forp-MFP(v, k) are obtained. For (p, k)=(3,3), (5,5), (2,3) and (3,4), necessary and sufficient conditions onv are obtained for the existence of ap-MFP(v, k). Further, for 3≤k≤5 and for any primep≡1 modk(k−1), we establish necessary and sufficient conditions onv for the existence of ap-MFP(v, k).     

On translation planes of orderq 3 that admit a collineation group of orderq 3

Let II be a translation plane of orderq ³, with kernel GF(q) forq a prime power, that admits a collineation groupG of orderq ³ in the linear translation complement. Moreover, assume thatG fixes a point at infinity, acts transitively on the remaining points at infinity andG/E is an abelian group of orderq ², whereE is the elation group ofG.In this article, we determined all such translation planes. They are (i) elusive planes of type I or II or (ii) desirable planes.Furthermore, we completely determined the translation planes of orderp ³, forp a prime, admitting a collineation groupG of orderp ³ in the translation complement such thatG fixes a point at infinity and acts transitively on the remaining points at infinity. They are (i) semifield planes of orderp ³ or (ii) the Sherk plane of order 27.     

On the generators of subgroups of unit groups of group rings

In this paper we find the generators of a subgroup of finite index in the unit group of the integral group ring of the metacyclic group of orderpq given byG=(a,x:a ^p=1=x ^q ,xax ⁻¹=a ^f ), wherep is an odd prime,q>2 a divisor ofp-1, andf belongs to the exponentq modulop.     

Simulated factorizations II

Summary A factorization of a finite abelian group is said to be simulated if it is obtained from a factorization into a direct product of subgroups by changing at mostk elements in each subgroup. The question has been asked as to which values ofk imply that in fact at least one subgroup must be left unaltered. This has been shown to be true fork = 1 but to be false, in general, fork = p – 1, wherep is the least prime dividing the order ofG. In this paper it is shown to be true fork = p – 2.     

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.     

Dimension subgroups and<Emphasis Type="Italic">p</Emphasis>-th powers in<Emphasis Type="Italic">p</Emphasis>-groups

We prove that if the nilpotence class of ap-group is strictly less thanp ^kthen every product ofp ^k-thpowers can be written as thep-th power of an element. Scoppola and Shalev have proven the same thing for groups of class strictly less thanp ^k−p ^k−1. They also provide an example which proves that ours is the best possible result. This is a generalization of the well known fact that in groups of class strictly less thanp every product ofp-powers is again ap-th power. Along the way we prove results of independent interest on dimension subgroups ofp-groups.     

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 nonabelian <Emphasis Type="Italic">p</Emphasis>-groups having exactly one maximal subgroup with a noncyclic center

We prove here that a nonabelian finite p-group G has exactly one maximal subgroup with a noncyclic center if and only if Z(G) is cyclic and G has exactly one normal abelian subgroup of type (p, p).     

Finite groups with trivial Frattini subgroup

All groups considered are finite. A group has a trivial Frattini subgroup if and only if every nontrivial normal subgroup has a proper supplement.The property is normal subgroup closed, but neither subgroup nor quotient closed. It is subgroup closed if and only if the group is elementary, i.e. all Sylow subgroups are elementary abelian. If G is solvable, then G and all its quotients have trivial Frattini subgroup if and only if every normal subgroup of G has a complement. For a nilpotent group, every nontrivial normal subgroup has a supplement if and only if the group is elementary abelian. Consequently, the center of a group in which every normal subgroup has a supplement is an elementary abelian direct factor.