On minimal irregular p-Groups

Brian Hartley asked me whether a free (nilpotent of class 2 and exponent p2)-group of countable rank has a faithful linear representation of finite degree, p here being a prime of course. The answer is yes. The point is that this then yields via work of F. Leinen and M. J. Tomkinson, see [3,3.6] an image of a linear p-group, which is not even finitary linear. The question of which relatively free groups have faithful linear representations dates back at least to work of W. Magnus in the 1930's, see [4, pp. 33, 34 and the final comment on p. 40] for a discussion of this. Our construction, which works more generally, is a further contribution. We write ℜc for the variety of nilpotent groups of class at most c and (ℭ9 for the variety of groups of exponent dividing q.

Powerfully nilpotent groups

The study of homology groups ${\mathfrak {B}_n}(\Pi ,A),\mathfrak {B}$ a variety, $\Pi$ a group in $\mathfrak {B}$, and $A$ a suitable $\Pi$-module, is continued. A ’Tor’ is constructed which gives a better (but imperfect) approximation to these groups than a Tor previously considered. ${\mathfrak {B}_2}(\Pi ,Z)$ is calculated for various varieties $\mathfrak {B}$ and groups $\Pi$.

The following results are established: (i) there exists a nonfinitely based variety of (nilpotent of class 2)-by-(nilpotent of class 2) groups, whose relatively free groups are torsion free; (ii) for every prime p, p≥3, there exists a nonfinitely based metanilpotent variety of groups of exponent p2.

Let G be a group and let x, y ∈ G. If every element of G can be written as a word $${x^{{\alpha _1}}}{y^{{\alpha _2}}}{x^{{\alpha _3}}} \ldots {x^{{\alpha _{n - 1}}}}{y^{{\alpha _n}}}$$ (1) where αi ∈ ℤ, 1 ≤ i ≤ n, then we say that x and y generate G and that G is a 2-generator group. Although cyclic groups are 2-generator groups according to this definition we are only interested here in 2-generator groups which cannot be generated by a single element. Even among finite groups G many are not 2-generator groups; for example the abelian group of order 8 in which every element has order 2 cannot be 2-generated since given any pair of distinct non-trivial elements x, y there are only 4 words given by expressions of the form (1). However, many groups are 2-generated, in particular finite simple groups.

In this article, we investigate the powerful nilpotency class of powerfully nilpotent groups of standard nilpotency class 2. We outline the process of collecting data using the computer algebra system GAP, formulating a conjecture based on the data, and finally we prove the conjecture. In particular, we prove that for a powerfully nilpotent group of nilpotency class 2 and order pn , where p is an odd prime, the powerful nilpotency class of G is at most the integer part of . We also identify and explain what this means in terms of the powerful coclass of the group.

We investigate the possible structures imposed on a finite group by its possession of an automorphism sending a large fraction of the group elements to their cubes, the philosophy being that this should force the group to be, in some sense, close to abelian. We prove two main theorems. In the first, we completely classify all finite groups with an automorphism cubing more than half their elements. All such groups are either nilpotent class 2 or possess an abelian subgroup of index 2. For our second theorem, we show that if a group possesses an automorphism sending more than 4/15 of its elements to their cubes, then it must be solvable. The group A_5 shows that this result is best-possible. Both our main findings closely parallel results of previous authors on finite groups possessing an automorphism which inverts many elements. The technicalities of the new proofs are somewhat more subtle, and also throw up a nice connection to a basic problem in combinatorial number theory, namely the study of subsets of finite cyclic groups which avoid non-trivial solutions to one or more translation invariant linear equations.

For a prime p, we denote by Bn the cyclic group of order pn. Let φ be a faithful irreducible character of Bn, where p is an odd prime. We study the p-group G containing Bn such that the induced character φG is also irreducible. The purpose of this article is to determine the subgroup NG(NG(Bn)) of G under the hypothesis [NG(Bn):Bn]4 ≦ pn.

By a nilpotent map we mean an orientably regular map whose orientation preserving automorphism group is nilpotent. The nilpotent maps are concluded to the maps whose automorphism group is a $2$-group and a complete classification of nilpotent maps of (nilpotency) class $2$ is given by Malnič et al. in [European J. Combin. 33 (2012), 1974-1986]. It is proved by Conder et al. in [J. Algebraic Combin. 44 (2016), 863-874] that given the class, there are finitely many simple nilpotent maps. However, for the nilpotent maps with multiple edges and given class, since its automorphism group may be infinitely big, it is impossible to list it by a computer. Therefore, to classify the nilpotent maps with small class $c$ is necessary and interesting. In this paper, the nilpotent maps of class $4$ will be determined.

We present a new algorithm which uses a cohomological approach to determine the groups of order pn, where p is a prime. We develop two methods to enumerate p-groups using the Cauchy-Frobenius Lemma. As an application we show that there are 10 494213 groups of order 29.1991 Mathematics subject classification (Amer. Math. Soc.): primary 20D15.

Abelian subgroups of order pm of the I-groups of the abelian groups of order pn and type 1,1,1,…

Finite group theorists have been interested in counting groups of prime-power order, as a preliminary step to counting groups of any finite order and to assist in explicitly listing such groups. In 1960, G. Higman considered when the functions giving the number of groups of prime-power order pn, for fixed n and varying p, is of a particular form, called polynomial on residue classes (PORC). The suggestion that such counting functions are PORC is known as Higman's PORC conjecture. In his 1960 paper [4] he proved that a certain class of groups of prime-power order, now called exponent-p class two groups, have counting functions that are PORC, but did not furnish explicit PORC functions.

We obtain sufficient conditions for the existence of a noninner automorphism of order p for finite p-groups. We show that groups of order pn (n 3) possess a noninner automorphism of order p.

1·1. We shall write for the class of groups all of whose non-trivial normal subgroups have finite index. Thus, rather obviously, finite groups, simple groups, and the infinite cyclic and dihedral groups all lie in the class . Other examples of -groups are to be found in a variety of contexts. The main result of Mennicke (8) shows that, for n ≥ 3, the factor group of SL(n, Z) by its centre is a -group, where SL(n, Z) denotes the unimodular group of all n × n invertible matrices with integer coefficients and with determinant 1. In McLain(7), an example is given of an infinite, locally finite, locally soluble -group, whose only non-trivial normal subgroups are the terms of its derived series, and in (4), P. Hall discusses an infinite -group, all of whose proper quotients are finite p-groups. If is a class of groups closed under taking homomorphic images, it is easily seen that the existence of an infinite -group satisfying the maximal condition for normal subgroups implies the existence of an infinite -group in the class , so that certain questions concerning the finiteness of groups satisfying the maximal condition for normal subgroups can be interpreted as questions about -groups.

The class of connected Labelled Oriented Graph (LOG) groups coincides with the class of fundamental groups of complements of closed, orientable 2-manifolds embedded inS4{S^{4}}, and so contains all knot groups. We investigate when Campbell and Robertson’s generalized Fibonacci groupsH⁢(r,n,s){H(r,n,s)}are connected LOG groups. In doing so, we use the theory of circulant matrices to calculate the Betti numbers of their abelianizations. We give an almost complete classification of the groupsH⁢(r,n,s){H(r,n,s)}that are connected LOG groups. All torus knot groups and the infinite cyclic group arise and we conjecture that these are the only possibilities. As a corollary we show thatH⁢(r,n,s){H(r,n,s)}is a 2-generator knot group if and only if it is a torus knot group.