P-groups Of Class Research Articles

Remarks on connections between the Leopoldt conjecture, p-class groups and unit groups of algebraic number fields

Remarks on connections between the Leopoldt conjecture, p-class groups and unit groups of algebraic number fields

On regularity of small primes in function fields

Let p be a finite prime of the rational function field K= F q ( T) and K( p ): K the pth cyclotomic extension. We study the p-components of various class groups associated with K( p ), using criteria of Kummer-Herbrand-Ribet type and explicit formulas for Bernoulli-Goss and Bernoulli-Carlitz numbers. A conjecture is formulated that hypothetically gives necessary and sufficient conditions for the non-vanishing of the p-class group of the ring of integers in the maximal “totally real” subfield K + ( p ) of K( p ).

The cyclic length of ap-group need not be logarithmic

The cyclic length l(G) of a finite supersolvable group G is defined to be the smallest number l such that there is a series G = N o > N I > . . . > N ~ = I , where No, N1 . . . . . Nl are l+1 different normal subgroups of G and all factor groups Ni_l/Ni (i=1 . . . . . l) are cyclic. L. R6dei asked the question whether or not the cyclic lengh has a logarithmic property on the class of p-groups: i.e. does l(G1 • G2) = =l(G1)+l(G2) hold, whenever Gx, G2 are p-groups for a certain prime p? At the first glance the answer could be expected to be in the positive, especially in view of the validity on the class of abelian p-groups. In the present note we will bring this expectation to nought not only by means of a single counterexample but more generally by assigning a p-group G to each abelian p-group H so that l ( G • < l ( G ) + l ( H ) takes place. The notation will be standard and can be found in [1]. All groups under consideration are finite.

Structure of semisimple differentials and p-class groups in ?p-extensions

For a wildly ramified p-extension E/F of algebraic function fields of one variable in an algebraically closed field k of characteristic p with Galois Group G, Nakajima obtained two exact sequences which determined implicitly the structure of the holomorphic semisimple differentials as k[G]-module. In this paper, in many cases, e.g., if there is a fully ramified prime, the structure is determined explicitly. Analogous results are obtained for p-extensions of ℤp-fields of CM-type. In the latter situation, if E/F is unramified, the structure of the minus part of the p-class group of E is determined as ℤp[G]-module.

Densities for Ranks of Certain Parts of p-Class Groups

Densities for Ranks of Certain Parts of p-Class Groups

Densities for ranks of certain parts of 𝑝-class groups

Let K K be a Galois extension of the field of rational numbers of prime degree p p , and let C K {C_K} be the p p -class group of K K . In this paper densities for the ranks of certain parts of such C K {C_K} are calculated, and these densities suggest a way to extend conjectures of Cohen and Lenstra.

Calculation of the Baer-invariant of certain groups

The present paper is devoted to calculation of the Baer-invariants of the dihedral and the quaternion groups of order 8, and also a certain class ofp-groups with respect to the variety of nilpotent groups.

An application of matrices over finite fields to algebraic number theory

This paper ultilizes properties of matrices over finite fields to obtain information about the rank of the p-class group of certain algebraic number fields.

On the Automorphism Group of a Finite p-Group with a Small Central Quotient

In recent years there has been considerable interest in the conjecture that |G| divides |Aut G| for all finite non-cyclic p-groups G of order greater than p2. In particular, the conjecture has been established for a considerable number of (not necessarily distinct) classes of finite p-groups ([6], [7], [8], [9], [15], [16]); additionally, results have been obtained, often using homological methods, which permit reductions in any attempt to establish the overall conjecture ([5], [10], [13], [15]). In the former case, the p-groups G have generally been regular p-groups (see, for example, [6]) and the prime p = 2 has either been excluded (see, for example, [8]) or treated as a special case (as in [9]).It is the purpose of this paper to establish the conjecture for the class of all p-groups G where |G: Z(G)| ≦ p4 with no restrictions on the prime p.

On the p-class groups of a Galois number field and its subfields

On the p-class groups of a Galois number field and its subfields

Conjugacy classes in finite groups

In the first part of this note, we give new proofs of known results regarding the class number of finite groups, adding a few related results. In the second part, we improve a result of Ito concerning a special class ofp-groups.

Irreducible Automorphisms of Certainp-Groups

The chief purpose of this paper is to find all pairs (G, θ) whereGis a finite specialp-group, andθis an automorphism ofGacting trivially on the Frattini subgroup and irreducibly on the Frattini quotient. This problem arises in the context of describing finite groups having an abelian maximal subgroup. In fact, we solve a more general problem for a wider class ofp-groups, which we callspecial F-groups,whereFis a finite field of characteristicp.We point out that ifpis odd, then anF-group has exponentp.On the other hand, every special 2-group is also a specialGF(2)-group.

Detachablep-groups and quasi-injectivity

If E is a ring, and G is an E-module, then G is quasi-injective if every homomorphism from a submodule of G to G can be extended to an endomorphism of G. POOLE and REID [2] raise the question as to which abelian groups G are quasiinjective as modules over their endomorphism rings E. They prove that direct sums of cyclic p-groups are quasi-injective. This result is extended here to a large class of p-groups, including all p-groups that have no elements of infinite height, and all totally projective p-groups -- in particular, all countable p-groups. The central idea is a property shared by totally projective p-groups and pgroups with no elements of infinite height. DEFINITION. If G is a p-group and xEG[p], then x is said to be detachable if G can be written as G1 @ Gz with (x) =p'G~ [p] for some ordinal c~. If every element of G[p] is detachable, we say that G is detachable. We write p'G~[p] instead of p~G~ to allow Ga to be divisible. Note that every element in G[p] of finite height is detachable, so every p-group with no elements of infinite height is detachable. Hill's extension of Ulm's theorem to totally projective p-groups (see [3]) implies that these groups are also detachable. THEOREM 1. If G is a detachable p-group, and x, yEG[p], then there is an endomorphism f of G such that f(x) =y or f(y) = x.

Two questions in the theory of formations

(1) If 9 is a saturated formation, let ,gS denote the formation of all finite soluble groups in which the S-projectors and g-normalizers coincide. By [2], 4.6, the formation gS is saturated if, and only if, gF = MS. In $1 we shall give the canonical local definition of the smallest saturated formation containing dU,. From this we shall deduce that if %F is saturated, 3 necessarily has the form 9 = Y,@(p). (2) A formation 9 is said to have the cover-avoidance property if the S-projectors of each group Q either cover or avoid each chief factor of (li. It was shown in [2], 5.9, that if a saturated formation 9 has the coveravoidance property, then either 9 = %Yz for some set T of primes or 3 = -9$F( p) where S(p) is a formation whose characteristic is the single prime p. We show in $2 that this second possibility can only occur when S(p) = YP , the class of p-groups. Combining these results we obtain the following conclusion: A saturated formation 9 has the cover-avoidance property if, and only if, 9 is either the formation of m-groups for some set z of primes or the formation of p-nilpotent groups for some prime p.

Finite groups with isomorphic group algebras

Introduction. Let G denote a finite group and F a field. The group algebra (or group ring) F(G) is the algebra over F with a basis multiplicatively isomorphic with G. Consider the following problem: If G is a group, F a field, find all groups H such that F(G) and F(H) are isomorphic over F. Perlis and Walker [9] solved this problem for Abelian groups over ordinary fields. Deskins [5] solved the problem for Abelian groups over modular fields. In this paper some partial results are obtained in connection with this problem for non-Abelian groups over an ordinary field. In ?1 we see some effects that certain subgroups of G and ideals of F(G) have on F(G). In ?2 a certain class of p-groups is studied. The theorem in ? 2 along with a theorem of S. D. Berman leads to an example of two distinct groups whose group algebras are isomorphic over any ordinary field. Group rings over the complex and real number fields and over the ring of integers are discussed in ? 3. Some examples, including the one mentioned above, are given in the last section. PRELIMINARY REMARKS. The following notations will be used. A (D B denotes the direct sum of algebras A and B over F. A ?F B =A ( B denotes the tensor product (direct product) of A and B over F. If K is an extension field of F, and if A is an algebra over F, then AK is the set K ?F A, considered as an algebra over K. G x H denotes the direct product of groups G and H.

On p-Groups of Class Three Generated by Three Elements

On p-Groups of Class Three Generated by Three Elements

On 𝑝-groups of class three generated by three elements

On 𝑝-groups of class three generated by three elements

Concerning Uniqueness-Bases of Finite Groups with Applications to p-Groups of Class 2

Concerning Uniqueness-Bases of Finite Groups with Applications to p-Groups of Class 2