Conjugate biprimitive finite groups saturated with finite simple subgroups

Coherence, subgroup separability, and metacyclic structures for a class of cyclically presented groups

We study periodic groups such that the normaliser of any finite non-trivial subgroup of such a group is almost layer-finite. The class of groups satisfying this condition is rather wide and includes the free Burnside groups of odd period which is greater than 665 and the groups constructed by A. Yu. Olshanskii.We consider the classical question: how the properties of the system of subgroups of a group influence on the properties of the group? We show that almost layer-finiteness is transferred on the group G from the normalisers of non-trivial finite subgroups of the group G if G is periodic conjugately biprimitively finite group with a strongly embedded subgroup.We study the structure of an infinite Sylow 2-subgroup in a periodic conjugately biprimitively finite group in the case that the normaliser of any finite non-trivial subgroup is almost layer-finite.The results of the paper can be useful in the study of the class of periodic conjugately biprimitively finite groups (periodic Shunkov groups).

Let Fr denote the free group of rank r and Out Fr: = AutFr/Inn Fr the outer automorphism group of Fr (automorphisms modulo inner automorphisms). In [10] we determined the maximal order 2rr! (for r > 2) for finite subgroups of Out Fr as well as the finite subgroup of that order which, for r > 3, is unique up to conjugation. In the present paper we determine all maximal finite subgroups (that is not contained in a larger finite subgroup) of Out F3, up to conjugation (Theorem 2 in Section 3). Here the considered case r = 3 serves as a model case: our method can be applied for other small values of r (in principle for any value of r) but the computations become considerably longer and are more apt for a computer then; the method can also be applied to determine the maximal finite subgroups of the automorphism group Aut Fr of Fr. Note that the canonical projection Aut Fr ⃗ Out Fr is injective on finite subgroups of Aut Fr; however, not all finite subgroups of Out Fr lift to finite subgroups of Aut Fr.

We show that an infinite residually finite boundedly generated group has an infinite chain of finite index subgroups with ranks uniformly bounded, and give (sublinear) upper bounds on the ranks of arbitrary finite index subgroups of boundedly generated groups (examples that come close to achieving these bounds are presented). This proves a strong form of a conjecture of Abert, Jaikin-Zapirain, and Nikolov that asserts that the rank gradient of infinite boundedly generated residually finite groups is zero. Furthermore, our first result establishes a variant of a conjecture of Lubotzky on the ranks of finite index subgroups of special linear groups over the integers, and is analogous to a result of Pyber and Segal for solvable groups.

Given an infinite group G, we consider the finitely additive invariant measure defined on finite unions of cosets of finite index subgroups. We show that this shares many properties with the size of subsets of a finite group, for instance we can obtain equivalent results on the Ruzsa distance and product free sets. In particular, if G has infinitely many finite index subgroups, then it has subsets S of measure arbitrarily close to 1/2 with square S2 having measure less than 1. We also examine properties of the Ruzsa distance on the set of finite index subgroups of an infinite group, whereupon it becomes a genuine metric.

Consider some finite group G and a finite subgroup H of G. Say that H is c-quasinormal in G if G has a quasinormal subgroup T such that HT = G and T ∩ H is quasinormal in G. Given a noncyclic Sylow subgroup P of G, we fix some subgroup D such that 1 < |D| < | P| and study the structure of G under the assumption that all subgroups H of P of the same order as D, having no supersolvable supplement in G, are c-quasinormal in G.

We show that given an affine algebraic group G over a field K and a finite subgroup scheme H of G there exists a finite dimensional G-module V such that V∣H is free. The problem is raised in the recent paper by Kuzucuŏglu and Zalesskiǐ [15] which contains a treatment of the special case in which K is the algebraic closure of a finite field and H is reduced. Our treatment is divided into two parts, according to whether K has zero or positive characteristic. The essence of the characteristic 0 case is a proof that, for given n, there exists a polynomial GLn(Q)-module V of dimension 2 n Π p p ( 2 n ) , where the product is over all primes less than or equal to n+1, such that V is free as a QH-module for every finite subgroup H of GLn(Q). The module V is the tensor product of the exterior algebra Λ*(E), on the natural GLn(Q)-module E, and Steinberg modules Stp, one for each prime not exceeding n+1. The Steinberg modules also play the major role in the case in which K has characteristic p>0 and the key point in our treatment is to show that for a finite subgroup scheme H of a general linear group scheme (or universal Chevalley group scheme) G over K, the Steinberg module St p n for G is injective (and projective) on restriction to H for n≫0. A curious consequence of this is that, despite the wild behaviour of the modular representation theory of finite groups, one has the following. Let H be a finite group and V a finite dimensional vector space. Then there exists a (well-understood) faithful rational representation π GL(V)→GL(W) such that, for each faithful representation ρ: H→GL(V), the composite πορ: H→GL(W) is free, in particular all representations πορ are equivalent.

The well-known characterization of two-ended groups says that every two-ended group can be split over finite subgroups which means it is isomorphic to either by a free product with amalgamation [Formula: see text] or an HNN-extension [Formula: see text], where [Formula: see text] is a finite group and [Formula: see text] and [Formula: see text]. In this paper, we show that there is a way in order to spilt two-ended quasi-transitive graphs without dominated ends and two-ended transitive graphs over finite subgraphs in the above sense. As an application of it, we characterize all groups acting with finitely many orbits almost freely on those graphs.

Involutory reflection groups and their models

In this paper we will determine all locally finite subgroups of a sfield and give a criterion for determining when the group algebra of an ascending solvable torsion free group has a quotient sfield. Notation and Definitions.A group G has property E if it can be embedded in a sfield D and property EE if every automorphism of the group G can be extended to be an automorphsim of D. For a more complete discussion see [5].The result on locally finite groups depends mainly on a result of Amitsur (see [l]).A complete discussion of the following notation can be found in his paper.k will denote the set of all primes and 7Ti the set of all odd primes p such that 2 has odd order mod p.Let m and r be relatively prime integers.Put s= (r -1, m), t = m/s and = minimal integer satisfying rn=l mod m.Denote by Gm,r a group generated by two elements A and B satisfying Am = \, Bn = Al and BAB~l = Ar.Denote by G", a group G which has a countable ascending tower of subgroups {H:0^< oo j such that G = ULi Hi and each Hi is isomorphic to Gmi,Ti.T*, 0*, I* will denote the binary tetrahedral, octahedral and icosahedral groups.Let p be a fixed prime dividing m. a = ap is the highest power of p dividing m. i]p is the minimal integer satisfying r^=\ mod (mp~a).pp is the minimal integer satisfying r" = " mod (mp~a) for some integer p'.Sj, is the minimal integer such that ps"=l mod (mp~a).o'P=pp8p/rp.Condition C. Integers m and r satisfy Condition C if either (I) (re, t) = (s, 0 = 1, or (II) re = 2re', m = 2am', s = 2s' where a^2, m', s' and re' are odd integers; (re, t) = (s, t)=2 and r= -1 mod 2".And either(III) n = 5 = 2 and r= -1 (mod m), or (IV) for every q/n there exists a prime p\m such that q\-qp and that either

Sharper applications to group theory are given of an elegant construction — the "tree-limit"—which S. Shelah circulated as a preprint in 1977 and used to obtain $\infty$-$\omega$-enlargements to power ${2^\omega }$ of certain countable homogeneous groups and skew fields. In this paper we enlarge the class of groups to which this construction can be interestingly applied and we obtain permutation representations of countable degree of the tree-limit groups; we obtain uncountable subgroup-incomparability for enlargements of countable existentially closed groups and even in nonhomogeneous cases we obtain the very strong "archetypal direct limit property" (which implies $\infty$-$\omega$-equivalence (see (1.0)) of the permutation representations). We are able to control the permutation representations which get stretched by the tree-limit by varying the point-stabilizer subgroups (see (5.5)). In particular we can archetypally stretch in ${2^\omega }$ subgroup-incomparable ways any homogeneous permutation representation of a countable locally finite group in which every finite subgroup has infinitely many regular orbits (Theorem 4). We discuss cases where tree-limits are subgroups of inverse limits.

Let G G be a finite group and let H H be a subgroup of G G . If g ∈ G g \in G , then the set E g = { h g h − 1 | h ∈ H } {E_g} = \{ hg{h^{ - 1}}|h \in H\} is the subclass of G G containing g g and Σ x ∈ E g x {\Sigma _{x \in {E_g}}}x is the subclass sum containing g g . The algebra over the field of complex numbers generated by these subclass sums is called the subclass algebra (denoted by S S ) associated with G G and H H . The irreducible modules of S S are demonstrated, and results about Schur algebras are used to develop formulas relating the irreducible characters of S S to the irreducible characters of G G and H H .

The lattices of eight- and ten-dimensional Euclidean space with irreducible automorphism group or, equivalently, the conjugacy classes of these groups in G L n ( Z ) \mathrm {GL}_n(\mathbb {Z}) for n = 8 , 10 n = 8,10 , are classified in this paper. The number of types is 52 in the case n = 8 n = 8 , and 47 in the case n = 10 n = 10 . As a consequence of this classification one has 26, resp. 46, conjugacy classes of maximal finite irreducible subgroups of G L 8 ( Z ) \mathrm {GL}_8(\mathbb {Z}) , resp. G L 10 ( Z ) \mathrm {GL}_{10}(\mathbb {Z}) . In particular, each such group is absolutely irreducible, and therefore each of the maximal finite groups of degree 8 turns up in earlier lists of classifications.

Finite Groups and Subgroups

A group is $\frac 32$-generated if every non-trivial element is part of a generating pair. In 2019, Donoven and Harper showed that many Thompson groups are $\frac 32$-generated and posed five questions. The first of these is whether there exists a 2-generated group with every proper quotient cyclic that is not $\frac 32$-generated. This is a natural question given the significant work in proving that no finite group has this property, but we show that there is such an infinite group. The groups we consider are a family of finite index subgroups $G_1,\, G_2,\, \ldots$ of the Houghton group $\operatorname {FSym}(\mathbb {Z})\rtimes \mathbb {Z}$. We then show that $G_1$ and $G_2$ are $\frac 32$-generated and investigate the related notion of spread for these groups. We are able to show that they have finite spread at least 2. These are, therefore, the first infinite groups to be shown to have finite positive spread, and the first to be shown to have spread at least 2 (other than $\mathbb {Z}$ and the Tarski monsters, which have infinite spread). As a consequence, for each $k\in \{2,\, 3,\, \ldots \}$, we also have that $G_{2k}$ is index $k$ in $G_2$ but $G_2$ is $\frac 32$-generated whereas $G_{2k}$ is not.