P-groups Of Class Research Articles

A generalization of Alperin Fusion theorem and its applications

Let F be a saturated fusion system on a finite p-group S, and let P be a strongly F-closed subgroup of S. We define the concept “F-essential subgroups with respect to P” which are some proper subgroups of P satisfying some technical conditions and show that an F-isomorphism between subgroups of P can be factorized by some automorphisms of P and F-essential subgroups with respect to P. When P is taken to be equal to S, the Alperin-Goldschmidt fusion theorem can be obtained as a special case. We also show that P⊴F if and only if there is no F-essential subgroup with respect to P. The following definition is made: A p-group P is strongly resistant in saturated fusion systems if P⊴F whenever there is an over p-group S and a saturated fusion system F on S such that P is strongly F-closed. It is shown that several classes of p-groups are strongly resistant, which appears as our third main theorem. We also give a new necessary and sufficient criterion for a strongly F-closed subgroup to be normal in F. These results are obtained as a consequence of developing a theory of quasi and semi-saturated fusion systems, which seems to be interesting in its own right.

The finitude of tamely ramified pro-p extensions of number fields with cyclic p-class groups

Let p be an odd prime and F be a number field whose p-class group is cyclic. Let F{q} be the maximal pro-p extension of F which is unramified outside a single non-p-adic prime ideal q of F. In this work, we study the finitude of the Galois group G{q}(F) of F{q} over F. We prove that G{q}(F) is finite for the majority of q's such that the generator rank of G{q}(F) is two, provided that for p=3, F is not a complex quartic field containing the primitive third roots of unity.

The Chevalley–Herbrand formula and the real abelian Main Conjecture (New criterion using capitulation of the class group)

Analytic theory of real abelian fields K says (in the cyclic semi-simple case) that the order of the p-class group HK is equal to the p-index of cyclotomic units (EK:FK). We develop, in this article, new promising links between: (i) the Chevalley–Herbrand formula giving the number of “ambiguous classes” in p-extensions L/K, L⊂K(μℓ) for auxiliary prime numbers ℓ≡1(mod2pN) inert in K; (ii) the phenomenon of capitulation of HK in L; (iii) the Main Theorem for the isotypic components using the irreducible p-adic characters φ of K, that we had conjectured (1977). We prove that this real Main Theorem is trivially fulfilled as soon as HK capitulates in L (Theorem 1.2). Computations with PARI programs support this new philosophy of the Main Theorem and the very frequent phenomenon of capitulation suggests Conjecture 1.1.

Orbit isomorphic skeleton groups

Recent development in the classification of p-groups often concentrate on the coclass graph G(p,r) associated with the finitep-groups coclassr, specially on periodicity results on these graphs. In particular, the structure of the subgraph inducedby ‘skeleton groups’ is of notable interest. Given their importance, in this paper, we investigate periodicity results of skeleton groups. Our results concentrate on the skeleton groups in G(p,1). We find a family of skeleton groups in G(7,1) whose 6-step parent is not aperiodic parent. This shows that the periodicity results available inthe current literature for primes p≡5 mod 6 do not hold for the primes p≡1 mod 6. We also improve a known periodicity result in a special case of skeleton groups.

On the λ-stability of p-class groups along cyclic p-towers of a number field

Let [Formula: see text] be a number field, [Formula: see text] a prime and [Formula: see text] a set of finite places of [Formula: see text]. We call [Formula: see text] a totally [Formula: see text]-ramified cyclic [Formula: see text]-tower if [Formula: see text] and if [Formula: see text] is totally ramified. Using analogues of Chevalley’s formula [G. Gras, Invariant generalized ideal classes—Structure theorems for p-class groups in p-extensions, Proc. Math. Sci. 127(1) (2017) 1–34], we give an elementary proof of a stability theorem (Theorem 3.1) for generalized [Formula: see text]-class groups [Formula: see text] of the layers [Formula: see text]: let [Formula: see text] given in Definition 1.1 ([Formula: see text] for ordinary class groups); then [Formula: see text] for all [Formula: see text], if and only if [Formula: see text]. This improves the case [Formula: see text] of [T. Fukuda, Remarks on [Formula: see text]-extensions of number fields, Proc. Japan Acad. Ser. A 70(8) (1994) 264–266; J. Li, Y. Ouyang, Y. Xu and S. Zhang, l-Class groups of fields in Kummer towers, Publ. Mat. 66(1) (2022) 235–267; Y. Mizusawa, K. Yamamoto, On [Formula: see text]-class groups of relative cyclic [Formula: see text]-extensions, Arch. Math. 117(3) (2021) 253–260] whose techniques are based on Iwasawa’s theory or Galois theory of pro-[Formula: see text]-groups. We deduce promising capitulation properties of [Formula: see text] in the tower giving Conjecture 4.1. Finally, we apply our principles to the torsion groups [Formula: see text] of abelian [Formula: see text]-ramification theory. Numerical examples are given with PARI programs.

Character degrees, character codegrees and nilpotence class of p-groups

Du and Lewis raised in 2016 the question of whether the nilpotence class of a p-group is bounded in terms of the number of character codegrees. In 2020, Croome and Lewis, gave a positive answer to this question for p-groups with four character codegree under some additional hypotheses related, for instance, to the number of character degrees of the group. In this note, we show that in general the nilpotence class of a p-group is bounded in terms of the number of character degrees and the number of character codegrees. In the case of four character codegrees, we extend some of the results of Croome and Lewis.

Algorithmic complexity of Greenberg’s conjecture

Let k be a totally real number field and p a prime. We show that the “complexity” of Greenberg’s conjecture ( $$\lambda = \mu = 0$$ ) is governed (under Leopoldt’s conjecture) by the finite torsion group $${{\mathscr {T}}}_k$$ of the Galois group of the maximal abelian p-ramified pro-p-extension of k, by means of images, in $${{\mathscr {T}}}_k$$ , of ideal norms from the layers $$k_n$$ of the cyclotomic tower (Theorem 4.2). These images are obtained via the algorithm computing, by “unscrewing”, the p-class group of $$k_n$$ . Conjecture 4.3 of equidistribution of these images would show that the number of steps $$b_n$$ of the algorithms is bounded as $$n \rightarrow \infty $$ , so that (Theorem 3.3) Greenberg’s conjecture, hopeless within the sole framework of Iwasawa’s theory, would hold true “with probability 1”.

Tate–Shafarevich groups in the cyclotomic [formula omitted]-extension and Weber's class number problem

Let K=Q(N) be the Nth layer in the cyclotomic Zˆ-extension. Many authors (Aoki, Fukuda, Horie, Ichimura, Inatomi, Komatsu, Miller, Morisawa, Nakajima, Okazaki, Washington,…) prove results on the p-class groups CK. We enlarge “Weber's problem” to the Tate–Shafarevich groups IIIK1≃CKSp[p] and IIIK2 having same p-rank as the more easily computable torsion group, TK, of the Galois group of the maximal abelian p-ramified pro-p-extension of K; but TK is often non-trivial, which raises questions for class groups since ▪, where RK is the normalized p-adic regulator. We give a new method testing TK≠1 (Theorem 4.6, Table in Appendix A.7) and characterize the fields K1=KQ(p) with CK1≠1 (Main Theorem 1.1 affirming, for short, that CK1≠1 if and only if p totally splits in K and TK≠1); this highlights the analytical results and justifies the eight known examples. All PARI/GP programs are given for further investigations.

On p-class groups of relative cyclic p-extensions

We prove a general stability theorem for $p$-class groups of number fields along relative cyclic extensions of degree $p^2$, which is a generalization of a finite-extension version of Fukuda's theorem by Li, Ouyang, Xu and Zhang. As an application, we give an example of pseudo-null Iwasawa module over a certain $2$-adic Lie extension.

Cohen–Lenstra Moments for Some Nonabelian Groups

Cohen and Lenstra detailed a heuristic for the distribution of odd p-class groups for imaginary quadratic fields. One such formulation of this distribution is that the expected number of surjections from the class group of an imaginary quadratic field k to a fixed odd abelian group is 1. Class field theory tells us that the class group is also the Galois group of the Hilbert class field, the maximal unramified abelian extension of k, so we could equivalently say the expected number of unramified G-extensions of k is 1/#Aut(G) for a fixed abelian group G. We generalize this to asking for the expected number of unramified G-extensions Galois over $\mathbb{Q}$ for a fixed finite group G, with no restrictions placed on G. We review cases where the answer is known or conjectured by Boston-Wood, Boston-Bush-Hajir, and Bhargava, then answer this question in several new cases. In particular, we show when the expected number is zero and give a nontrivial family of groups realizing this. Additionally, we prove the expected number for the quaternion group $Q_8$ and dihedral group $D_4$ of order 8 is infinite. Lastly, we discuss the special case of groups generated by elements of order 2 and give an argument for an infinite expected number based on Malle's conjecture.

Finite p-Groups and Coclass Theory

Symmetries arise in many areas of science, and group theory, as the Mathematics of Symmetry, models this natural concept. A prominent theme in group theory is the classification of p-groups since these groups play an important role in group theory. However, the sheer number of p-groups of a fixed size makes such a classification impossible; a different and fruitful approach is to use coclass. This project makes further advances in coclass theory, providing new results for skeleton groups and space groups.

Genus theory and ε-conjectures on p-class groups

We suspect that the “genus part” of the class number of a number field K may be an obstruction for an “easy and unconditional proof” of the classical p-rank ε-conjecture for p-class groups and, a fortiori, for a proof of the “strong ε-conjecture”: #(CℓK⊗Zp)≪d,p,ε(|DK|)ε for all K of degree d. We analyze the weight of genus theory in this inequality by means of an infinite family of degree p cyclic fields with many ramified primes, then we prove the p-rank ε-conjecture: #(CℓK⊗Fp)≪d,p,ε(|DK|)ε, for d=p and the family of degree p cyclic extensions (Theorem 2.5) then sketch the case of arbitrary base fields. The possible obstruction for the strong form, in the degree p cyclic case, is the order of magnitude of the set of “exceptional” p-classes given by a well-known non-predictible algorithm, but largely controlled thanks to recent density results due to Koymans–Pagano. Then we compare the ε-conjectures with some p-adic conjectures, of Brauer–Siegel type, about the torsion group TK of the Galois group of the maximal abelian p-ramified pro-p-extension of totally real number fields K. We give numerical computations with the corresponding PARI/GP programs.

On the $p$-rank of the ideal class group of a normal extension with simple Galois group

Let p be a prime and L a finite normal extension over a number field K whose Galois group is simple and non-abelian. The aim of this paper is to estimate a lower bound of the ratio of the p-rank of the ideal class group of L to the p-rank of the ambiguous p-class group of L with respect to K.

Maximal non-commuting set in 2-generated p-groups of class two

Let [Formula: see text] be a group. A set [Formula: see text] in [Formula: see text] is said to be non-commuting if [Formula: see text] for any two distinct elements [Formula: see text] and [Formula: see text] in [Formula: see text]. If [Formula: see text] for any other non-commuting set [Formula: see text] in [Formula: see text], then [Formula: see text] is said to be a maximal non-commuting set. In this paper, we obtain the cardinality of a maximal non-commuting set in [Formula: see text]-generated [Formula: see text]-groups of class two having finite commutator subgroup.

Capitulation in the cyclotomic ℤ2 extension of CM number fields

AbstractLet 𝕂n be the intermediate steps in the cyclotomic ℤp-extension of a CM number field 𝕂. For p ⧧ 2 the minus part of the p-class groups is given by An− = $\frac{1}{2}$(1 − j)An. We will give a new definition of the minus part for p = 2 and will prove that there is no finite submodule in lim∞←nAn−. Furthermore we will show that μ = 0 if and only if μ− = 0 in this new definition.

On the Invariant Factors of Class Groups in Towers of Number Fields

AbstractFor a finite abelian p-group A of rank d = dim A/pA, let A := be its (logarithmic) mean exponent. We study the behavior of themean exponent of p-class groups in pro-p towers L/K of number fields. Via a combination of results from analytic and algebraic number theory, we construct infinite tamely ramified pro-p towers in which the mean exponent of p-class groups remains bounded. Several explicit examples are given with p = 2. Turning to group theory, we introduce an invariant attached to a finitely generated pro-p group G; when G = Gal(L/K), where L is the Hilbert p-class field tower of a number field K, measures the asymptotic behavior of the mean exponent of p-class groups inside L/K. We compare and contrast the behavior of this invariant in analytic versus non-analytic groups. We exploit the interplay of group-theoretical and number-theoretical perspectives on this invariant and explore some open questions that arise as a result, which may be of independent interest in group theory.

Automorphism groups of finitely generated groups that are isomorphic to group of inner automorphisms

Let G be a finitely generated group and let C* denote the group of all central automorphisms of G fixing the center of G elementwise. Azhdari and Malayeri [J. Algebra Appl. 6 (2011), 1283–1290] gave necessary and sufficient conditions on G such that C* ≃ Inn(G). We prove a more general technical lemma and, as a consequence, obtain a short and easy proof of this result of Azhdari and Malayeri and generalize their result; and also generalize the main result of Nasrabadi and Farimani [Indag. Math. (N. S.) 26 (2015), 137–141] from finite autonilpotent p-groups of class 2 to arbitrary finitely generated nilpotent groups of class 2. Subsequently, we also obtain short proofs of some other existing and some new related results.

Non-triviality of Tate cohomology for certain classes of finite p-groups

ABSTRACTWe prove that the Tate cohomology groups Ĥn(G∕Φ(G),Z(Φ(G))) are non-trivial, whenever G is a finite p-group of class 3, or the pth term of the upper central series of G contains Z(Φ(G)). This confirms a conjecture of Schmid for these groups.

Invariant generalized ideal classes – structure theorems for p-class groups in p-extensions

We give, in Sections 2 and 3, an english translation of: {\it Classes g\'en\'eralis\'ees invariantes}, J. Math. Soc. Japan, 46, 3 (1994), with some improvements and with notations and definitions in accordance with our book: {\it Class Field Theory: from theory to practice}, SMM, Springer-Verlag, $2^{\rm nd}$ corrected printing 2005. We recall, in Section 4, some structure theorems for finite $\mathbb{Z}_p[G]$-modules ($G \simeq \mathbb{Z}/p\,\mathbb{Z}$) obtained in: {\it Sur les $\ell$-classes d'id\'eaux dans les extensions cycliques relatives de degr\'e premier $\ell$}, Annales de l'Institut Fourier, 23, 3 (1973). Then we recall the algorithm of local normic computations which allows to obtain the order and (potentially) the structure of a $p$-class group in a cyclic extension of degree $p$. In Section 5, we apply this to the study of the structure of relative $p$-class groups of Abelian extensions of prime to $p$ degree, using the Thaine--Ribet--Mazur--Wiles--Kolyvagin "principal theorem", and the notion of "admissible sets of prime numbers" in a cyclic extension of degree $p$, from: {\it Sur la structure des groupes de classes relatives}, Annales de l'Institut Fourier, 43, 1 (1993). In conclusion, we suggest the study, in the same spirit, of some deep invariants attached to the $p$-ramification theory (as dual form of non-ramification theory) and which have become standard in a $p$-adic framework. Since some of these techniques have often been rediscovered, we give a substantial (but certainly incomplete) bibliography which may be used to have a broad view on the subject.

Criteria for Three-Stage Towers of <i>p</i>-Class Fields

Let p be a prime and K be a number field with non-trivial p-class group ClpK. A crucial step in identifying the Galois group G∞p of the maximal unramified pro-p extension of K is to determine its two-stage approximation M=G2pk, that is the second derived quotient MsG/Gn. The family τ1K of abelian type invariants of the p-class groups ClpL of all unramified cyclic extensions L/K of degree p is called the index- abelianization data (IPAD) of K. It is able to specify a finite batch of contestants for the second p-class group M of K. In this paper we introduce two different kinds of generalized IPADs for obtaining more sophisticated results. The multi-layered IPAD (τ1Kτ(2)K) includes data on unramified abelian extensions L/K of degree p2 and enables sharper bounds for the order of M in the case Clpks(p,p,p), where current im-plementations of the p-group generation algorithm fail to produce explicit contestants for M , due to memory limitations. The iterated IPAD of second order τ(2)K contains information on non-abelian unramified extensions L/K of degree p2, or even p3, and admits the identification of the p-class tower group G for various infinite series of quadratic fields K=Q(√d) with ClpKs(p,p) possessing a p-class field tower of exact length lpK=3 as a striking novelty.