Automorphism Groups of Infinite Semilinear Orders (I)

All groups considered are finite. A group has a trivial Frattini subgroup if and only if every nontrivial normal subgroup has a proper supplement.The property is normal subgroup closed, but neither subgroup nor quotient closed. It is subgroup closed if and only if the group is elementary, i.e. all Sylow subgroups are elementary abelian. If G is solvable, then G and all its quotients have trivial Frattini subgroup if and only if every normal subgroup of G has a complement. For a nilpotent group, every nontrivial normal subgroup has a supplement if and only if the group is elementary abelian. Consequently, the center of a group in which every normal subgroup has a supplement is an elementary abelian direct factor.

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.

To search for nontrivial proper normal subgroups is often the first step in the investigation of a finite group. For example, if the group G has such a normal subgroup N, then in proofs by induction one frequently gets information about N and G/N, allowing one to derive the desired result for G (e.g., 6.1.2 on page 122).

This article began as a study of the structure of infinite permutation groups $G$ in which point stabilisers are finite and all infinite normal subgroups are transitive. That led to two variations. One is the generalisation in which point stabilisers are merely assumed to satisfy min-n, the minimal condition on normal subgroups. The groups $G$ are then of two kinds. Either they have a maximal finite normal subgroup, modulo which they have either one or two minimal nontrivial normal subgroups, or they have a regular normal subgroup $M$ which is a divisible abelian $p$-group of finite rank. In the latter case the point stabilisers are finite and act irreducibly on a $p$-adic vector space associated with $M$. This leads to our second variation, which is a study of the finite linear groups that can arise.

We describe the structure of virtually solvable normal subgroups in the automorphism group of a right-angled Artin group Aut ⁢ ( A Γ ) \mathrm{Aut}(A_{\Gamma}) . In particular, we prove that a finite normal subgroup in Aut ⁢ ( A Γ ) \mathrm{Aut}(A_{\Gamma}) has at most order two and if Γ is not a clique, then any finite normal subgroup in Aut ⁢ ( A Γ ) \mathrm{Aut}(A_{\Gamma}) is trivial. This property has implications for automatic continuity and C ∗ C^{\ast} -algebras: every algebraic epimorphism φ : L ↠ Aut ⁢ ( A Γ ) \varphi\colon L\twoheadrightarrow\mathrm{Aut}(A_{\Gamma}) from a locally compact Hausdorff group 𝐿 is continuous if and only if A Γ A_{\Gamma} is not isomorphic to Z n \mathbb{Z}^{n} for any n ≥ 1 n\geq 1 . Furthermore, if Γ is not a join and contains at least two vertices, then the set of invertible elements is dense in the reduced group C ∗ C^{\ast} -algebra of Aut ⁢ ( A Γ ) \mathrm{Aut}(A_{\Gamma}) . We obtain similar results for Aut ⁢ ( G Γ ) \mathrm{Aut}(G_{\Gamma}) , where G Γ G_{\Gamma} is a graph product of cyclic groups. Moreover, we give a description of the center of Aut ⁢ ( G Γ ) \mathrm{Aut}(G_{\Gamma}) in terms of the defining graph Γ.

g*: For each normal subgroup N$ 'P(G), each reduced product of G over N is a semidirect product. (G = NB is a reduced product over a normal subgroup N by a subgroup B iff B does not contain a proper subgroup B* such that G = NB*.) F. Gross [5] has shown that for a finite solvable group G having 4!(G) = 1, splitting over each normal subgroup is sufficient for the subgroup lattice to be comple

Product graphs for given subgroups of the wreath product of two groups, II

We compare Aut(Q), the classical automorphism group of a countable dense linear order, with Autc(Q), the group of all computable automorphisms of such an order. They have a number of similarities, including the facts that every element of each group is a commutator and each group has exactly three nontrivial normal subgroups. However, the standard proofs of these facts in Aut(Q) do not work for Autc(Q). Also, Aut(Q) has three fundamental properties which fail in Autc(Q): it is divisible, every element is a commutator of itself with some other element, and two elements are conjugate if and only if they have isomorphic orbital structures.

A set D of 3-transpositions in the group G is a normal set of elements of order 2 such that, for all d and e in D, the order of the product de is 1, 2, or 3. If G is generated by the conjugacy class D of 3-transpositions, we say that (G, D) is a 3-transposition group or (loosely) that G is a 3-transposition group. The study of 3-transposition groups was instituted by Bernd Fischer [6, 7, 8] who classified all 3-transposition groups which are finite and have no non-trivial normal solvable subgroups. Recently the present author and H. Cuypers[5] extended Fischer's result to include all 3-transposition groups with trivial centre. For this classification the present paper provides the extension of Fischer's paper [8] where he gave two basic reductions, the Normal Subgroup Theorem and the Transitivity Theorem stated below. Other results of help in the classification are also presented here.

We prove that if a countable discrete group Γ is w-rigid, i.e. it contains an infinite normal subgroup H with the relative property (T) (e.g. $\Gamma=SL(2,\mathbb{Z})\ltimes\mathbb{Z}^2$ , or Γ=H×H’ with H an infinite Kazhdan group and H’ arbitrary), and $\mathcal{V}$ is a closed subgroup of the group of unitaries of a finite separable von Neumann algebra (e.g. $\mathcal{V}$ countable discrete, or separable compact), then any $\mathcal{V}$ -valued measurable cocycle for a measure preserving action $\Gamma\curvearrowright X$ of Γ on a probability space (X,μ) which is weak mixing on H and s-malleable (e.g. the Bernoulli action $\Gamma\curvearrowright[0,1]^{\Gamma}$ ) is cohomologous to a group morphism of Γ into $\mathcal{V}$ . We use the case $\mathcal{V}$ discrete of this result to prove that if in addition Γ has no non-trivial finite normal subgroups then any orbit equivalence between $\Gamma\curvearrowright X$ and a free ergodic measure preserving action of a countable group Λ is implemented by a conjugacy of the actions, with respect to some group isomorphism Γ≃Λ.

Symmetric cubic graphs with non-solvable automorphism groups

1·1. A group is called characteristically simple if it has no proper non-trivial subgroups which are left invariant by all of its automorphisms. One familiar class of characteristically simple groups consists of all direct powers of simple groups: this contains all finite characteristically simple groups, and, more generally, all characteristically simple groups having minimal normal subgroups. However not all characteristically simple groups lie in this class because, for instance, additive groups of fields are characteristically simple. Our object here is to construct finitely generated groups, and also groups satisfying the maximal condition for normal subgroups, which are characteristically simple but which are not direct powers of simple groups.

The problem of locating and classifying the normal subgroups of GLn(R), the general linear group of degree n over a commutative ring R with an identity element, has received considerable attention. The solution when R is a field is well known (of. Dieudonné(5), Artin(1)): unless n is equal to two and R has two or three elements, normal subgroups of GLn(R) either lie in the centre of GLn(R) or contain the special linear group SLn(R). However, if R is not a field, then for each ideal I of R the natural map R → R/I induces a homomorphismand, if 0 < I < R, the kernel of θ1 is a non-central normal subgroup of GLn(R) which does not contain SLn(R). The most that may be expected is that each normal subgroup determines an ideal I of R, in such a way that all normal subgroups determining the same ideal I lie between suitably defined greatest and smallest normal subgroups of GLn(R) corresponding to I. For example, write ZI for the inverse image of the centre of GLn(R/I) under the homomorphism θI, and write KI for the intersection of SLn(R) with the kernel of θI. Then KI ≽ ZI, and the results of Klingenberg(7) and Mennicke(9) show that if n ≥ 3, and if R is either a local ring or the ring of rational integers, then any normal subgroup H of GLn(R) satisfiesfor some uniquely determined ideal I. There are many similar theorems. That the above result breaks down for arbitrary rings, even for large n, follows easily from the negative solution to the congruence subgroup problem for certain rings of algebraic integers (see Bass, Milnor and Serre(3)).

Schreier proved in [8] tha t a finitely generated normal subgroup U # { 1 } of a free group F is of finite index. This result was extended by Karrass and Solitar in [3], to the case when U is not necessarily normal, but contains a non-trivial normal subgroup of F. In Topology, the free groups occur as fundamental groups of surfaces with boundary, and we here extend the result still further (Theorem 6.1) to the case when F is the (non-abelian) fundamental group of any connected surface, with or without boundary, except for a Klein bottle. We use topological methods, and also the elements of Morse theory, although the latter could be ehminated. A sketch of this theory is included, however, par t ly for its intuitive appeal, and part ly because the Morse theory picks out stable generators of the fundamental group, and therefore is helpful as a tool. Indeed, the author was able to use it quickly to prove Schreier's Theorem, and Theorem 3.3 below (that an open surface has free fundamental group), before knowing tha t proofs already existed in the literature. Our exposition is always from the point of view tha t it is the surface, ra ther than the group, which is the ult imate object o f study; and we have perhaps laboured points tha t might irk the pure group-theorist.

We associate with every étale groupoid$\mathfrak{G}$two normal subgroups$\mathsf{S}(\mathfrak{G})$and$\mathsf{A}(\mathfrak{G})$of the topological full group of$\mathfrak{G}$, which are analogs of the symmetric and alternating groups. We prove that if$\mathfrak{G}$is a minimal groupoid of germs (e.g., of a group action), then$\mathsf{A}(\mathfrak{G})$is simple and is contained in every non-trivial normal subgroup of the full group. We show that if$\mathfrak{G}$is expansive (e.g., is the groupoid of germs of an expansive action of a group), then$\mathsf{A}(\mathfrak{G})$is finitely generated. We also show that$\mathsf{S}(\mathfrak{G})/\mathsf{A}(\mathfrak{G})$is a quotient of$H_{0}(\mathfrak{G},\mathbb{Z}/2\mathbb{Z})$.