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}.     

<Emphasis Type="Italic">p</Emphasis>-groups having few almost-rational irreducible characters

We prove that a 2-group has exactly five rational irreducible characters if and only if it is dihedral, semidihedral or generalized quaternion. For an arbitrary prime p, we say that an irreducible character χ of a p-group G is “almost rational” if ℚ(χ) is contained in the cyclotomic field ℚ _p , and we write ar(G) to denote the number of almost-rational irreducible characters of G. For noncyclic p-groups, the two smallest possible values for ar(G) are p ² and p ² + p − 1, and we study p-groups G for which ar(G) is one of these two numbers. If ar(G) = p ² + p − 1, we say that G is “exceptional”. We show that for exceptional groups, |G: G′| = p ², and so the assertion about 2-groups with which we began follows from this. We show also that for each prime p, there are exceptional p-groups of arbitrarily large order, and for p ≥ 5, there is a pro-p-group with the property that all of its finite homomorphic images of order at least p ³ are exceptional.     

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.     

The intersection of subgroups of finite <Emphasis Type="Italic">p</Emphasis>-groups

For a finite p-group G and a positive integer k let I _k (G) denote the intersection of all subgroups of G of order p ^k . This paper classifies the finite p-groups G with I_k(G) @ C_p^k-1{{I}_k(G)\cong C_{p^{k-1}}} for primes p > 2. We also show that for any k, α ≥ 0 with 2(α + 1) ≤ k ≤ n−α the groups G of order p ⁿ with I_k(G) @ C_p^k-a{{I}_k(G)\cong C_{p^{k-\alpha}}} are exactly the groups of exponent p ^n-α .     

Conjugacy classes and finite p-groups

Let G be a finite p-group, where p is a prime number, and a ∈ G. Denote by Cl(a) = {gag⁻¹| g ∈ G} the conjugacy class of a in G. Assume that |Cl(a)| = pⁿ. Then Cl(a) Cl(a⁻¹) = {xy | x ∈ Cl(a), y ∈ Cl(a⁻¹)} is the union of at least n(p − 1) + 1 distinct conjugacy classes of G. Received: 16 December 2004     

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.     

Finite <Emphasis Type="Italic">p</Emphasis>-central groups of height <Emphasis Type="Italic">k</Emphasis>

A finite group G is called p ⁱ -central of height k if every element of order p ⁱ of G is contained in the k ^th -term ζ _k (G) of the ascending central series of G. If p is odd, such a group has to be p-nilpotent (Thm. A). Finite p-central p-groups of height p − 2 can be seen as the dual analogue of finite potent p-groups, i.e., for such a finite p-group P the group P/Ω₁(P) is also p-central of height p − 2 (Thm. B). In such a group P, the index of P ^p is less than or equal to the order of the subgroup Ω₁(P) (Thm. C). If the Sylow p-subgroup P of a finite group G is p-central of height p − 1, p odd, and N _G (P) is p-nilpotent, then G is also p-nilpotent (Thm. D). Moreover, if G is a p-soluble finite group, p odd, and P ∈ Syl _p (G) is p-central of height p − 2, then N _G (P) controls p-fusion in G (Thm. E). It is well-known that the last two properties hold for Swan groups (see [11]).     

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.     

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.     

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.     

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.     

Restriction of characters and products of characters

Let G be a finite p-group, for some prime p, and ψ, θ ∈ Irr(G) be irreducible complex characters of G. It has been proved that if, in addition, ψ and θ are faithful characters, then the product ψθ is a multiple of an irreducible or it is the nontrivial linear combination of at least (p + 1)/2 distinct irreducible characters of G. We show that if we do not require the characters to be faithful, then given any integer k > 0, we can always find a p-group P and irreducible characters Ψ and Θ of P such that the product ΨΘ is the nontrivial combination of exactly k distinct irreducible characters. We do this by translating examples of decompositions of restrictions of characters into decompositions of products of characters.     

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.     

Isomorphic digraphs from powers modulo <Emphasis Type="Italic">p</Emphasis>

Let p be a prime. We assign to each positive number k a digraph G _p ^k whose set of vertices is {1, 2, …, p − 1} and there exists a directed edge from a vertex a to a vertex b if a ^k ≡ b (mod p). In this paper we obtain a necessary and sufficient condition for G_p^k₁ @ G_p^k₂G_p^{{k_1}} \simeq G_p^{{k_2}}.     

Elementary Abelian <Emphasis Type="Italic">p</Emphasis>-groups of rank greater than or equal to 4<Emphasis Type="Italic">p</Emphasis>−2 are not CI-groups

In this paper we prove that an elementary Abelian p-group of rank 4p−2 is not a CI⁽²⁾-group, i.e. there exists a 2-closed transitive permutation group containing two non-conjugate regular elementary Abelian p-subgroups of rank 4p−2, see Hirasaka and Muzychuk (J. Comb. Theory Ser. A 94(2), 339–362, 2001). It was shown in Hirasaka and Muzychuk (loc cit) and Muzychuk (Discrete Math. 264(1–3), 167–185, 2003) that this is related to the problem of determining whether an elementary Abelian p-group of rank n is a CI-group. As a strengthening of this result we prove that an elementary Abelian p-group E of rank greater or equal to 4p−2 is not a CI-group, i.e. there exist two isomorphic Cayley digraphs over E whose corresponding connection sets are not conjugate in Aut E. This research was supported by a fellowship from the Pacific Institute for the Mathematical Sciences.     

Endomorphism algebras of transitive permutation modules for p-groups

Let G be a finite p-group with subgroup H and k a field of characteristic p. We study the endomorphism algebra E = End_kG(k_H ↑^G), showing that it is a split extension of a nilpotent ideal by the group algebra kN_G(H)/H. We identify the space of endomorphisms that factor through a projective kG-module and hence the endomorphism ring of k_H ↑^G in the stable module category, and determine the Loewy structure of E when G has nilpotency class 2 and [G, H] is cyclic. Received: 3 November 2008     

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

Let G be a pro-p group and let k ≥ 1. If γ _k(p−1) (G) ≤ γ _r for some r and s such that k(p − 1) < r + s(p − 1), we prove that the exponent of Ω_i(G) is at most p ^i+k−1 for all i. Supported by the Spanish Ministry of Science and Education, grant MTM2004-04665, partly with FEDER funds. The first author is also supported by the University of the Basque Country, grant UPV05/99. The second author is also supported by the Basque Government.     

Additive Latin transversals and group rings

LetA={a ₁, …,a _k} and {b ₁, …,b _k} be two subsets of an abelian groupG, k≤|G|. Snevily conjectured that, when |G| is odd, there is a numbering of the elements ofB such thata _i+b _i,1≤i≤k are pairwise distinct. By using a polynomial method, Alon affirmed this conjecture for |G| prime, even whenA is a sequence ofk<|G| elements. With a new application of the polynomial method, Dasgupta, Károlyi, Serra and Szegedy extended Alon’s result to the groupsZ _p ^r andZ _p ^rin the casek<p and verified Snevily’s conjecture for every cyclic group. In this paper, by employing group rings as a tool, we prove that Alon’s result is true for any finite abelianp-group withk<√2p, and verify Snevily’s conjecture for every abelian group of odd order in the casek<√p, wherep is the smallest prime divisor of |G|. This work has been supported partly by NSFC grant number 19971058 and 10271080.     

Elementary proof of a theorem of Blackburn

For a positive integer n, a finite p-group G is called an ℳ _n -group, if all subgroups of index p ⁿ of G are metacyclic, but there is at least one subgroup of index p ⁿ⁻¹ that is not. A classical result in p-group theory is the classification of ℳ₁-groups by Blackburn. In this paper, we give a slightly shorter and more elementary proof of this result.     

Degrees of irreducible characters of the suzuki 2-groups

The degrees of the irreducible characters of the Suzuki 2-groupsA(m,ϕ) are described. Assume that the order of an automorphism ϕ of the fieldGF (2 ^m ) isk>1,G=A(m,ϕ), and cd (G) is the set of degrees of the irreducible characters ofG. Ifk is odd, then cd(G)={1, 2( ^m−m/k)/2 }, and ifk=2, then cd(G)={1, 2 ^m/2 }. Ifk is even andk≠2 then cd(G)={1,2 ^m/2 , 2 ^m/2−m/k }, and the groupG has (2 ^m −1)2 ^m/k /(2 ^m/k +1) characters of degree2 ^m/2 and (2^m−1)2^2m/k/(2^m/k+1) characters of degree 2^m/2−m/k. Translated fromMatematicheskie Zametki, Vol. 66, No. 2, pp. 258–263, August, 1999.