Finite groups of order 2a3b13c (original) (raw)

3-local characterization of Held groups

Algebra and Logic, 1980

In the recent past a series of strong results have been announced, which essentially constitute an exhaustive treatment of the problem of describing the p -local structure of finite groups p of type characteristic two in the case where the G -rank ( p an odd prime), of the 2-local subgroups of G is sufficiently big (viz°, ~ ). The situation is much less clear in the case of small fl -rank. Here, it seems that a characterization would be useful of known simple groups, not necessarily of type characteristic two, by means of the centralizers of elements of order /D, or -in the first place -by means of the centralizers of elements of order three.

Locally finite groups and their subgroups with small centralizers

Journal of Algebra, 2017

Let p be a prime and G a locally finite group containing an elementary abelian p-subgroup A of rank at least 3 such that C G (A) is Chernikov and C G (a) involves no infinite simple groups for any a ∈ A #. We show that G is almost locally soluble (Theorem 1.1). The key step in the proof is the following characterization of P SL p (k): An infinite simple locally finite group G admits an elementary abelian p-group of automorphisms A such that C G (A) is Chernikov and C G (A) involves no infinite simple groups for any a ∈ A # if and only if G is isomorphic to P SL p (k) for some locally finite field k of characteristic different from p and A has order p 2. 2000 Mathematics Subject Classification. 20F50, 20E36.

A constructive approach: From local subgroups to new classes of finite groups

Hacettepe journal of mathematics and statistics, 2022

Let G be a finite group and S be a proper subgroup of G. A group G is called an S-(Squasinormal)-group if every local subgroup of G is either an S-quasinormal subgroup or conjugate to a subgroup of S. The main purpose of this construction is to demonstrate a new way of analyzing the structure of a finite group by the properties and the number of conjugacy classes of its local subgroups.

A Group Theoretical Characterisation of S-Arithmetic Groups in Higher Rank Semi-Simple Groups

2002

We now state the main theorem proved in the present paper. There are a number of definitions which are used in the statement. They will be explained in sections 1 and 2. Main Theorem. Let Γ be a group satisfying the following conditions. (1) Γ is finitely generated and is virtually torsion free (see the end of the introduction for an explanation of this term). (2) The semi-simple dimension of Γ (denoted s.s. dim (Γ)) is finite and is positive (see sections (1.1)-(1.3) for an explanation of the terms semisimple p−dimension (denoted s.s.p-dim)) and semi-simple dimension (denoted s.s.dim). (3) The semi-simple rank of Γ (denoted s..s. rank (Γ)) is at least two : s.s. rank (Γ) ≥ 2 (see (2.1) for the definition). (4) Γ is hereditarily just infinite (i.e. if Γ ⊂ Γ is a subgroup of finite index and N ⊂ Γ is a normal subgroup of Γ , then either Γ /N is finite or N is finite). (5) Let X def = {l | l is a prime and s.s.l-dim(Γ) < s.s.dim (Γ)} (we prove, under the assumptions (1) through (4), that X is a finite set: see the proof of Theorem (1.17)). Let q be a prime with q not in X. Form the product Ω = ∈X Γ × Γ q. Let Q l (l ∈ X) be the semi-simple quotient of Γ l and Q q that of Γ q (see (1.17) for a proof that under the assumption (2), such quotients exist). Let Ω = l∈X Q l × Q q ; there is a natural map from Γ to Ω which under the assumptions (1)-(4), is a virtual inclusion. Let Γ ⊂ Λ ⊂ Ω and Λ satisfy the properties (1), (2), (3) (but not necessarily (4)) as well as the equalities s.s.p − dim(Λ) = s.s.p − dim(Γ) for every prime p. Then we assume that Λ/Γ is finite. Under these assumptions (1) through (5), Γ is virtually isomorphic to an S−arithmetic subgroup Γ of a group G, which is defined over a number field K and is absolutely simple (here, S is a finite set of places of K containing all the Archimedean ones). Moreover, S−rank (G) def = v∈S K v −rank (G) ≥ 2, Γ ⊂ G(O S) , O S = S-integers on K, and G(O S)/Γ is finite. Remark: We note (see §1 and §2) that all these assumptions (s.s. dimension being finite, semi-simple rank being at least two, and the condition (5)) are properties of a quotient of a certain prop completion of Γ, and as such, are not really dependent on a linear realisation of Γ. However, it turns out (see Theorem (1.17)) that there is a more or less canonical linear realisation of our group Γ under these assumptions. Corollary 1. Suppose Γ is a group satisfying the properties (1) through (5) of the Main Theorem. Suppose that s.s.p−dim(Γ) = s.s.dim(Γ) for every prime p (in other words, suppose that the set X of (5) of the Main Theorem is empty). Then, Γ is isomorphic to an arithmetic subgroup of a higher rank semisimple real Lie group. K such that S−rank (G) := v∈S K v −rank (G) ≥ 2. Let Γ be a subgroup of finite index in G(O s) (O S = S-integers in K). Then Γ satisfies the propertis (1) through (5). It is of interest to note that in fact, we can recover-purely from the abstract properties of Γ-the S-dimension of the semi-simple group in which Γ sits as a lattice , and also its S-rank. However, the field K over which our group G is defined, and the set S of places, are irrecoverable by our methods (our methods use the prop completion of Γ and there are non-isomorphic arithmetic groups with isomorphic profinite completions (see section (3.1)). We prove the Main Theorem in §3. In the course of the proof, we also use Theorem 2. Among the S-arithmetic groups characterised by the Main Theorem, it would be desirable to single out the non-uniform ones. In §3 we also define the notion of a u-element (following [L-M-R1]) and deduce Corollary 3. Let Γ be an abstract group possessing the properties (1) through (5) of the Main Theorem. Suppose Γ has a u−element of infinite order. Then Γ is isomorphic to an S−arithmetic non-uniform lattice.