John Poland

The number of conjugacy classes of non-normal subgroups in nilpotent groups

In a recent paper, Rolf Brandi classified all finite groups having exactly one conjugacy class of nonnormal subgroups, and conjectured thatfor a nilpotent group G of nilpotency class c = c(G) the number v(G) = vof conjugacy classes of nonnormal subgroups satisfies the inequality v(G) ≥ c(G) – 1 (with the exception of the Hamiltonian groups, of course). The purpose of this paper is to establish this conjecture and to decide when this inequality is sharp.

Profinite completions and isomorphic finite quotients

Let G be a group and {G,} be the set of all normal subgroups of finite index in G. Then the set {G/G,, Gob} of finite quotients G/G, of G together with the canonical projections gob : G/G, ---) G/Gb whenever G, c Gb is an inverse system. The inverse limit_ I@ G/G, of this system is called the profinite completion of G and is denoted by G. The group d can also be described as the closure of the image of G under the diagonal mapping d : GA n(G/G,) where G/G, is given the discrete topology and n(G/G,) has the product topology. In this description the elements of 6 are the elements (g,) E n (G/G,) which satisfy @&go) =gb whenever G, c Gb. The object of this paper is to prove the following result:

Elementary equivalence and the commutator subgroup

If G and H are elementarily equivalent groups (that is, no elementary statement of group theory distinguishes between G and H ) then the definable subgroups of G are elementarily equivalent to the corresponding ones of H . But G ′ of G , consisting of all products of commutators [a, b] = a −1 b −1 ab of elements a and b of G , may not be definable. Must G ′ and H ′ be elementarily equivalent?

On lattice isomorphic of mixed abelian groups

1. Introduction. In this paper, we investigate when two mixed nonsplitting abelian groups of torsion free rank one are lattice isomorphic. That is, their lattice of subgroups are isomorphic. This is the only outstanding case in the quest to know when abelian groups are lattice isomorphic, as we described in [4]. In [4], we also derived the necessary and sufficient conditions for mixed splitting abelian groups of torsion free rank one to be lattice isomorphic. Ostendorf has obtained a similar result in [6]. These conditions are that the torsion subgroups be isomorphic and the type of one of the groups can be obtained from the type of other by a permutation of the primes, that fixes primes occurring as orders of elements. This generalizes well-known results on when two splitting abelian groups of torsion free rank one are isomorphic. In the 1960s Rotman [8] and Megibben [5] proved that for many classes of mixed abelian groups G of torsion free rank one (eg : countable G) the height matrix U (G) was the distinguishing characteristic. It is our purpose here to extend our results in [4] to obtain parallel generalization of Rotman and Megibbens work. The necessary and sufficient conditions are summarized in the following two Theorems. Theorem I. Let G be a mixed abelian group of torsion free rank one, and assume that G is lattice isomorphic to an abelian group H. Then; H is a mixed abelian group of torsion free rank one, T(G) is isomorphic to T(H), and U(H) can be obtained from U(G) by permutation 11 of primes, with primes occurring as orders of elements of T(G) ~- T(H) fixed. Theorem II. Let G and H be two mixed abelian groups of torsion free rank one with T(G) ~- T(H). And assume that U (H) can be obtained from U (G) by a permutation 17 of primes, which fixes the primes occurring as orders of elements of T(G) ~- T(H). Now if; (i) G and H are splitting over their torsion subgroup, or (ii) G and 1t are countable, or (iii) T(G) ~ T(H) is a direct sum of countable groups, or (iv) T(G) ~ T(H) are both closed, then G is lattice isomorphic to H. We conclude this paper with an example of two mixed abelian groups of torsion free rank one with the same torsion parts and the same height matrices which are not lattice

The Shuffle Algorithm and Jordan Blocks

Abstract A shuffle is the horizontal interchange of a pair of blocks of the same size in a matrix. A general algorithm using row reduction and shuffles was first introduced by Luenberger, and then used by Anstreicher and Rothblum to give an algorithm to compute generalized nullspaces. We present a new, concise proof of this shuffle algorithm, and show how the shuffle algorithm can be used in deriving the Jordan blocks for a square matrix with known eigenvalues.