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.     

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.     

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.     

A class of special p-groups

Starting by the multiplication table of a finite group G, for each odd prime p a special p-group P of exponent p is constructed. Some connections between the structures of G and P, concerning subgroups and automorphisms, are given. When p does not divide the order of G, for every normal subgroup of G a direct decomposition of P is done. Besides, the order of the derived subgroup of G is proved to be connected with the existence of some abelian subgroups of maximal order in P. Finally, the elements of P with large centralizers are characterized in terms of algebraic properties of the multiplication table of G. Dedicated to Herman Heineken on the occasion of his 60th birthday Work supported by M.U.R.S.T. and G.N.S.A.G.A-C.N.R. of Italy.     

Finite p-groups not characteristic in any p-group in which they are properly contained

Answering a question raised by Y. Berkovich, we give examples of finite p-groups G with the property that the only finite p-group K with G char K, is G itself. We also prove a theorem stating that every finite p-group is contained in such a group G.     

A characterization of finite symplectic polar spaces of odd prime order

A sufficient condition for the representation group for a nonabelian representation (Definition 1.1) of a finite partial linear space to be a finite p-group is given (Theorem 2.9). We characterize finite symplectic polar spaces of rank r at least two and of odd prime order p as the only finite polar spaces of rank at least two and of prime order admitting nonabelian representations. The representation group of such a polar space is an extraspecial p-group of order p1+2r and of exponent p (Theorems 1.5 and 1.6).     

On automorphisms of a class of special p-groups

Let G be a finite group of order n, for some n\geqq 1 n\geqq 1 , and p be an odd prime number. In [5] Verardi has constructed a special p-group P_G P_G of exponent p such that |P_G|=p³ⁿ |P_G|=p^{3n} . In this paper, we calculate the order of Aut(P_G) (P_G) and prove that Aut(P_G) (P_G) is the semidirect product of two subgroups.     

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.     

On Finite p-Groups Whose Central Automorphisms are all Class Preserving

We obtain certain results on a finite p-group whose central automorphisms are all class preserving. In particular, we prove that if G is a finite p-group whose central automorphisms are all class preserving, then d(G) is even, where d(G) denotes the number of elements in any minimal generating set for G. As an application of these results, we obtain some results regarding finite p-groups whose automorphisms are all class preserving.     

Smooth Groups

A group is called smooth if it has a finite maximal chain of subgroups in which any two intervals of the same length are isomorphic (as lattices). We show that every finite smooth group G is a semidirect product of a p-group by a cyclic group; in particular, G is soluble. We determine the exact structure of G if G is not a p-group.     

Essential cohomology of finite groups

In this paper we prove that a finite group G with Cohen-Macaulay mod p cohomology will have non-trivial undetectable elements in if and only if G is a p-group such that every element of order p in G is central. Applications and examples are also provided. Received: April 18, 1996     

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.     

Quasinormal subgroups of order p 2

The location of quasinormal subgroups in a group is not particularly well known. Maximal ones always have to be normal, but little has been proved about the minimal ones. In finite groups, the difficulties arise in the p-groups. Here we prove that, for every odd prime p, a quasinormal subgroup of order p ² in a finite p-group G contains a quasinormal subgroup of G of order p. S. Stonehewer is grateful to the Australian National University for financial support during the preparation of this paper.     

P. Hall's covering group and embeddings of countable cc-groups

Let A be an elementary abelian group of order at least p ³ acting on a finite p′-group G that is soluble with derived length d. Assume that γ _c (C _G (a)) has exponent dividing m for any a ∈ A ^#. It is proved that there exist {p, d, c, m}-bounded numbers c ₁ and m ₁ such that γ _{c 1} (G) has exponent dividing m ₁.     

Minimal Number of Generators and Minimum Order of a Non-Abelian Group Whose Elements Commute with Their Endomorphic Images

A group in which every element commutes with its endomorphic images is called an “E-group″. If p is a prime number, a p-group G which is an E-group is called a “pE-group″. Every abelian group is obviously an E-group. We prove that every 2-generator E-group is abelian and that all 3-generator E-groups are nilpotent of class at most 2. It is also proved that every infinite 3-generator E-group is abelian. We conjecture that every finite 3-generator E-group should be abelian. Moreover, we show that the minimum order of a non-abelian pE-group is p ⁸ for any odd prime number p and this order is 2⁷ for p = 2. Some of these results are proved for a class wider than the class of E-groups.     

Some Properties of Certain Subgroups of Tensor Squares of p-Groups

Let G be a finite p-group of order p ⁿ and ?(G) be the subgroup of the tensor square of G generated by all symbols x ? x, for all x in G. In the present article, we construct an upper bound for the order of ?(G) and any extra special p-group. It is also shown that ?(G) ? ?(G/G′). Using our result, we obtain the explicit structure of the tensor square of G and π₃ SK(G, 1). Finally, the structure of G will be characterized when the bound is attained.     

The automorphism group of a nonsplit metacyclic p-group

This note considers a finite group G = HK, which is a product of a subgroup H and a normal subgroup K, and determines subgroups of Aut G. The special case when G is a nonsplit metacyclic p-group, where p is odd, is then considered and the structure of its automorphism group Aut G is given. Received: 13 September 2007, Revised: 22 November 2007     

Power automorphisms of finitep-groups

For a finite groupG letA(G) denote the group of power automorphisms, i.e. automorphisms normalizing every subgroup ofG. IfG is ap-group of class at mostp, the structure ofA (G) is shown to be rather restricted, generalizing a result of Cooper ([2]). The existence of nontrivial power automorphisms, however, seems to impose restrictions on thep-groupG itself. It is proved that the nilpotence class of a metabelianp-group of exponentp ² possessing a nontrival power automorphism is bounded by a function ofp. The “nicer” the automorphism—the lower the bound for the class. Therefore a “type” for power automorphisms is introduced. Several examples ofp-groups having large power automorphism groups are given.