Two-ended quasi-transitive graphs

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

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.

A group G is saturated with groups of the set X if every finite subgroup K≤G is embedded in G into a subgroup L isomorphic to some group of X. We study periodic biprimitive finite groups saturated with groups of the sets {L2(pn)}, {Re(32n+1)}, and {Sz(22n+1)}. It is proved thai such groups are all isomorphic to {L2(P)}, {Re(Q)}, or {Sr(Q)} over locally finite fields.

This chapter continues the development of group theory begun in Chapter IV, the main topics being the use of generators and relations, representation theory for finite groups, and group extensions. Representation theory uses linear algebra and inner-product spaces in an essential way, and a structure-theory theorem for finite groups is obtained as a consequence. Group extensions introduce the subject of cohomology of groups.Sections 1–3 concern generators and relations. The context for generators and relations is that of a free group on the set of generators, and the relations indicate passage to a quotient of this free group by a normal subgroup. Section 1 constructs free groups in terms of words built from an alphabet and shows that free groups are characterized by a certain universal mapping property. This universal mapping property implies that any group may be defined by generators and relations. Computations with free groups are aided by the fact that two reduced words yield the same element of a free group if and only if the reduced words are identical. Section 2 obtains the Nielsen-Schreier Theorem that subgroups of free groups are free. Section 3 enlarges the construction of free groups to the notion of the free product of an arbitrary set of groups. Free product is what coproduct is for the category of groups; free groups themselves may be regarded as free products of copies of the integers.Sections 4–5 introduce representation theory for finite groups and give an example of an important application whose statement lies outside representation theory. Section 4 contains various results giving an analysis of the space C(G, ?) of all complex-valued functions on a finite group G. In this analysis those functions that are constant on conjugacy classes are shown to be linear combinations of the characters of the irreducible representations. Section 5 proves Burnside’s Theorem as an application of this theory—that any finite group of order paqb with p and q prime and with a+b > 1 has a nontrivial normal subgroup.Section 6 introduces cohomology of groups in connection with group extensions. If N is to be a normal subgroup of G and Q is to be isomorphic to G/N, the first question is to parametrize the possibilities for G up to isomorphism. A second question is to parametrize the possibilities for G if G is to be a semidirect product of N and Q.

We consider finite groups which admit a faithful, smooth action on an acyclic manifold of dimension three, four or five (e.g. euclidean space). Our first main result states that a finite group acting on an acyclic 3- or 4-manifold is isomorphic to a subgroup of the orthogonal group O(3) or O(4), respectively. The analogue remains open in dimension five (where it is not true for arbitrary continuous actions, however). We prove that the only finite nonabelian simple groups admitting a smooth action on an acyclic 5-manifold are the alternating groups A_5 and A_6, and deduce from this a short list of finite groups, closely related to the finite subgroups of SO(5), which are the candidates for orientation-preserving actions on acyclic 5-manifolds.

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.

Given nontrivial finite groups $A$ and $B$, not both of order 2, we prove that every finite simple group of sufficiently large rank is an image of the free product $A \ast B$. To show this, we prove that every finite simple group of sufficiently large rank is generated by a pair of subgroups isomorphic to $A$ and $B$. This proves a conjecture of Tamburini and Wilson.

Let G be a group. An element g of G is called conjugacy distinguished or c.d. in G if and only if given any element h of G either h is conjugate to g or there is a homomorphism ξ \xi from G onto a finite group such that ξ ( h ) \xi (h) and ξ ( g ) \xi (g) are not conjugate in ξ ( G ) \xi (G) . Following A. Mostowski, a group G is conjugacy separable or c.s. if and only if every element of G is c.d. in G. In this paper we prove that every element conjugate to a cyclically reduced element of length greater than 1 in the free product of two free groups with a cyclic amalgamated subgroup is c.d. We also prove that a group formed by adding a root of an element to a free group is c.s.

We show that any free product of two (non-trivial) countable groups, one of them being infinite, admits a faithful and homogeneous action on the random graph. We also show that a large class of HNN extensions or free products, amalgamated over a finite group, admit such an action and we extend our results to groups acting on trees. Finally, we show the ubiquity of finitely generated free dense subgroups of the automorphism group of the random graph whose action on it have all orbits infinite.

Involutory reflection groups and their models

We show that the class of groups which have monoid presentations by means of finite special [λ]-confluent string-rewriting systems strictly contains the class of plain groups (the groups which are free products of a finitely generated free group and finitely many finite groups), and that any group which has an infinite cyclic central subgroup can be presented by such a string-rewriting system if and only if it is the direct product of an infinite cyclic group and a finite cyclic group.

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 &gt; 2) for finite subgroups of Out Fr as well as the finite subgroup of that order which, for r &gt; 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.

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.