On some Λ-analytic pro-p groups

This paper is devoted to the first steps towards a systematic study of pro-p groups which are analytic over a commutative Noetherian local pro-p ring Λ, e.g. Λ= . We restrict our attention to Λ-standard groups, which are pro-p groups arising from a formal group defined over Λ. Under some additional assumptions we show that these groups are of ‘intermediate growth’ in various senses, strictly betweenp-adic analytic pro-p groups and free pro-p groups. This suggests a refinement of Lazard's theory which stresses the dichotomy betweenp-adic analytic pro-p groups and all the others. In the course of the discussion we answer a question posed in [LM1], and settle two conjectures from [Bo].     

Discrete groups of slow subgroup growth

It is known that the subgroup growth of finitely generated linear groups is either polynomial or at least $n^{\frac{{\log n}}{{\log \log n}}} $ . In this paper we prove the existence of a finitely generated group whose subgroup growth is of type $n^{\frac{{\log n}}{{(\log \log n)^2 }}} $ . This is the slowest non-polynomial subgroup growth obtained so far for finitely generated groups. The subgroup growth typen ^logn is also realized. The proofs involve analysis of the subgroup structure of finite alternating groups and finite simple groups in general. For example, we show there is an absolute constantc such that, ifT is any finite simple group, thenT has at mostn ^{c logn} subgroups of indexn.     

The length and depth of algebraic groups

Mathematische Zeitschrift - Let G be a connected algebraic group. An unrefinable chain of G is a chain of subgroups $$G = G_0> G_1> \cdots > G_t = 1$$ , where each $$G_i$$ is a...     

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.     

Simple groups,maximal subgroups,and probabilistic aspects of profinite groups

We show that a finite simple group has at mostn ^1.875+o(1) maximal subgroups of indexn. This enables us to characterise profinite groups which are generated with positive probability by boundedly many random elements. It turns out that these groups are exactly those having polynomial maximal subgroup growth. Related results are also established.     

All-pay auctions with risk-averse players

We study independent private-value all-pay auctions with risk-averse players. We show that: (1) Players with low values bid lower and players with high values bid higher than they would bid in the risk neutral case. (2) Players with low values bid lower and players with high values bid higher than they would bid in a first-price auction. (3) Players’ expected utilities in an all-pay auction are lower than in a first-price auction. We also use perturbation analysis to calculate explicit approximations of the equilibrium strategies of risk-averse players and the seller’s expected revenue. In particular, we show that in all-pay auctions the seller’s expected payoff in the risk-averse case may be either higher or lower than in the risk neutral case.     

The Fixity of Groups of Prime-Power Order

It is well known that the only groups of prime-power order whichcan act fixed point freely on a complex linear space are thecyclic or generalised quaternion groups. Given a positive integerf and a prime p exceeding f, we determine the p-groups whichhave a faithful complex representation such that the dimensionsof the spaces of fixed points of non-trivial elements are atmost f.     

On the degree of groups of polynomial subgroup growth

Let be a finitely generated residually finite group and let denote the number of index subgroups of . If for some and for all , then is said to have polynomial subgroup growth (PSG, for short). The degree of is then defined by .

Very little seems to be known about the relation between and the algebraic structure of . We derive a formula for computing the degree of certain metabelian groups, which serves as a main tool in this paper. Addressing a problem posed by Lubotzky, we also show that if is a finite index subgroup, then .

A large part of the paper is devoted to the structure of groups of small degree. We show that is bounded above by a linear function of if and only if is virtually cyclic. We then determine all groups of degree less than , and reveal some connections with plane crystallographic groups. It follows from our results that the degree of a finitely generated group cannot lie in the open interval .

Our methods are largely number-theoretic, and density theorems à la Chebotarev play essential role in the proofs. Most of the results also rely implicitly on the Classification of Finite Simple Groups.