A p-nilpotency criterion

Let G be a finite group, p a fixed prime and P a Sylow p-subgroup of G. In this short note we prove that if p is odd, then G is p-nilpotent if and only if P controls fusion of cyclic groups of order p. For the case p = 2, we show that G is p-nilpotent if and only if P controls fusion of cyclic groups of order 2 and 4.     

On Two Theorems of Finite Solvable Groups

For a finite group G, let T（G） denote a set of primes such that a prime p belongs to T（G） if and only if p is a divisor of the index of some maximal subgroup of G. It is proved that if G satisfies any one of the following conditions： （1） G has a p-complement for each p∈T（G）; （2）│T（G）│= 2： （3） the normalizer of a Sylow p-subgroup of G has prime power index for each odd prime p∈T（G）; then G either is solvable or G/Sol（G）≌PSL（2, 7） where Sol（G） is the largest solvable normal subgroup of G.     

Virtually Nilpotent Groups with (almost) All Localizations Trivial

A group G is generically trivial if and only if, for all prime numbers p the localization of G with respect to p is trivial. Taking off from a theorem of Casacuberta and Castellet , we prove that a virtually nilpotent group E is generically trivial if and only if E is perfect. Inspired by this result, we introduce the concept of almost generically trivial groups. Those are groups G such that, for only finitely many primes p the localization of G with respect to p is not trivial. We prove that a virtually nilpotent group E with finitely generated abelianization is almost generically trivial if and only if the abelianization of E is finite.     

Bounding an Index by the Largest Character Degree of a p-Solvable Group

In this article, we show that if p is a prime and G is a p-solvable group, then |G: O _p (G)| _p  ≤ (b(G) ^p /p)^1/(p?1), where b(G) is the largest character degree of G. If p is an odd prime that is not a Mersenne prime or if the nilpotence class of a Sylow p-subgroup of G is at most p, then |G: O _p (G)| _p  ≤ b(G).     

On Commuting Automorphisms of Finite p-Groups with a Metacyclic Quotient

Let G be a finite non-Abelian p-group, where p is an odd prime, such that G/Z(G) is metacyclic. We prove that all commuting automorphisms of G form a subgroup of Aut(G) if and only if G is of nilpotence class 2.

     

On purifiable subgroups in arbitrary abelian groups

We determine the structure of a p-pure[pure] hull of a p-purifiable [purifiable] subgroup of an arbitrary abelian group. Moreover, we prove that a subgroup A of an abelian group G is purifiable in G if and only if A is p-purifiable in G for every prime p. Using these results, we characterize the groups G for which all subgroups are purifiable in G. Furthermore, we establish several properties of purifiable subgroups.     

Finite nonabelian <Emphasis Type="Italic">p</Emphasis>-groups having exactly one maximal subgroup with a noncyclic center

We prove here that a nonabelian finite p-group G has exactly one maximal subgroup with a noncyclic center if and only if Z(G) is cyclic and G has exactly one normal abelian subgroup of type (p, p).     

On the lattice formed by all normal subloops of a finite loop

All normal subloops of a loopG form a modular latticeL _n (G). It is shown that a finite loopG has a complemented normal subloop lattice if and only ifG is a direct product of simple subloops. In particular,L _n (G) is a Boolean algebra if and only if no two isomorphic factors occurring in a decomposition ofG are abelian groups. The normal subloop lattice of a finite loop is a projective geometry if and only ifG is an elementary abelianp-group for some primep.     

The K-theory of p-compact groups

In this paper, we show that the p-adic K-theory of a connected p-compact is the ring of invariants of the Weyl group action on the K-theory of a maximal torus. We apply this result to show that a connected finite loop space admits a maximal torus if and only if its complex K-theory is -isomorphic to the K-theory of some BG, where G is a compact connected Lie group. Received: November 9, 1996     

Connected components of the category of elementary abelian p-subgroups

We determine the maximal number of conjugacy classes of maximal elementary abelian subgroups of rank 2 in a finite p-group G, for an odd prime p. Namely, it is p if G has rank at least 3 and it is p+1 if G has rank 2. More precisely, if G has rank 2, there are exactly 1,2,p+1, or possibly 3 classes for some 3-groups of maximal nilpotency class.     

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.     

Choosability of Powers of Circuits

 It is proved that ch(G)=χ(G) if G=C _n ^p , the pth power of the circuit graph C _n , or if G is a uniform inflation of such a graph. The proof uses the method of Alon and Tarsi. As a corollary, the (a : b)-choosability conjectures hold for all such graphs. Received: October 10, 2000 Final version received: November 8, 2001     

Simple Twisted Group Algebras of Dimension p4 and their Semi-Centers

For a simple twisted group algebra over a group G, if G^∣ is Hall subgroup of G, then the semi-center is simple. Simple twisted group algebras correspond to groups of central type. We classify all groups of central type of order p⁴ where p is prime and use this to show that for odd primes p there exists a unique group G of order p⁴, such that there exists simple twisted group algebra over G with a commutative semi-center. Moreover, if 1 < |G| <64, then the semi-center of simple twisted group algebras over G is noncommutative and this bounds are strict.     

The Symmetric Genus of p-Groups

Let G be a finite group. The symmetric genus σ(G) is the minimum genus of any Riemann surface on which G acts. We show that a non-cyclic p-group G has symmetric genus not congruent to 1(mod p ³) if and only if G is in one of 10 families of groups. The genus formula for each of these 10 families of groups is determined. A consequence of this classification is that almost all positive integers that are the genus of a p-group are congruent to 1(mod p ³). Finally, the integers that occur as the symmetric genus of a p-group with Frattini-class 2 have density zero in the positive integers.     

<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.     

Bounded Direction–Length Frameworks

A direction–length framework is a pair (G,p) where G=(V;D,L) is a ‘mixed’ graph whose edges are labelled as ‘direction’ or ‘length’ edges and p is a map from V to ℝ ^d for some d. The label of an edge uv represents a direction or length constraint between p(u) and p(v). Let G ⁺ be obtained from G by adding, for each length edge e of G, a direction edge with the same end vertices as e. We show that (G,p) is bounded if and only if (G ⁺,p) is infinitesimally rigid. This gives a characterization of when (G,p) is bounded in terms of the rank of the rigidity matrix of (G ⁺,p). We use this to characterize when a mixed graph is generically bounded in ℝ ^d . As an application we deduce that if (G,p) is a globally rigid generic framework with at least two length edges and e is a length edge of G then (G∖e,p) is bounded.     

Groups whose vanishing class sizes are not divisible by a given prime

Let G be a finite group. An element ${g\in G}Let G be a finite group. An element g ? G{g\in G} is a vanishing element of G if there exists an irreducible complex character χ of G such that χ(g) = 0: if this is the case, we say that the conjugacy class of g in G is a vanishing conjugacy class of G. In this paper we show that, if the size of every vanishing conjugacy class of G is not divisible by a given prime number p, then G has a normal p-complement and abelian Sylow p-subgroups.