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.
A characterization of the simple group He
Journal of Algebra, 1979
The purpose of this paper is to prove the THEOREM. Let G be a jinite, noaz-abelian, simple group containing a 3-central element r of order 3 such that C,(rr)j(w> is isomorphic to A,. If G is not 3-~o~~a~ then G is isomorphic to He. This yields the following COKQLLARY. Let G be a jkite, non-abelian, sim@e group containing alz element T of order 3 such that Co(~))/(~) is isomorphic to A, or S,. Assume f~~~the~mQ~e at there exists in G an elementary abelian subgroup of order 9 all rzon-identity elements of which me conjugate to rr. Then G is isomorphic to He. The proof of the Theorem will be based on the following YPOTI-IESIS. Let G be a simple group containing a standard subgroup A stich A/Z(A) is isomorphic to PSL(3, 4). Then G : ZF i.~omoqphic to He or SW. OY Ol$r~ Here He, Suz, ON denote the sporadic simple groups of orders 1P33527~17, 21337527.1H "13, 2g345721 1.19.31 respectively discovered by O'Nan. In [3] Gheng proves the Hypothesis under one of the following further conditions: (i) The Sylow 2-subgroup of Z(A) is non-trivial. (ii) 211 does not divide the order of G. The full result might be established by now. The notation is standard (see [q and [lo]). In particular A, and S, denote respectively the alternating and symmetric group of degree n. PSL(n, q) is the projective special linear group of dimension n on a field with p elements. Z, denotes the cyclic group of order n and E,n the elementary abelian p-group of 261
Simple groups of order p · 3a · 2b
Journal of Algebra, 1970
In this paper some local group theoretic properties of a simple group G of order p * 3@ * 2b are found. These are applied in a later paper to show there are no simple groups of order 7 * 3" * 2b other than the three well-known ones. R. Brauer [4] has shown there are only the three known simple groups LI, , A, , and O,(3) of order 5 .3" * 2b. His treatment uses modular character theory especially for the prime 5. It follows from J. Thompson's X-group paper [9] that if G is a simple group of order p + 3" * 2*, then p = 5,7, 13, or 17. In our later treatment of the case p '=-T 7 we seem to need the results of the N-group paper itself. This present paper prepares the way.
On finite factorizable groups*1
Journal of Algebra, 1984
ON FINITE FACTORIZABLE GROUPS 523 (I) A, with r > 5 a prime and A N A,-, . (II) M,, and either A is solvable or A N M,,. (III) M,, and either B is Frobenius of order 11 . 23 or B is cyclic of order 23 and A N M,, .
Journal of Group Theory, 2007
A finite group G is said to be a PST-group if, for subgroups H and K of G with H Sylow-permutable in K and K Sylow-permutable in G, it is always the case that H is Sylowpermutable in G. A group G is a T *-group if, for subgroups H and K of G with H normal in K and K normal in G, it is always the case that H is Sylow-permutable in G. In this paper, we show that finite PST-groups and finite T *-groups are one and the same. A new characterisation of soluble PST-groups is also presented.
On some metabelian 3-groups and applications I
Gulf Journal of Mathematics, 2016
Let G be a 3-class group of maximal class, and γ 2 (G) = [G, G] its derived group. Assume that the commutator factor group G/γ 2 (G) is of type (3, 3) and the transfers , and, H is one of its maximal normals subgroups different to χ 2 (G). Then G is completely determined with the isomorphism class groups of maximal class defined by B.Nebelung in . Moreover the group G is realised. At the end numerical examples illustrating the results are given.
Bulletin of Mathematical Sciences, 2012
The notion of adequate subgroups was introduced by Thorne (J Inst Math Jussieu. arXiv:1107.5989, to appear). It is a weakening of the notion of big subgroup used by Wiles and Taylor in proving automorphy lifting theorems for certain Galois representations. Using this idea, Thorne was able to prove some new lifting theorems. It was shown in Guralnick et al. (J Inst Math Jussieu. arXiv:1107.5993, to appear) that certain groups were adequate. One of the key aspects was the question of whether the span of the semisimple elements in the group is the full endomorphism ring of an absolutely irreducible module. We show that this is the case in prime characteristic p for p-solvable groups as long the dimension is not divisible by p. We also observe that the condition holds for certain infinite groups. Finally, we present the first examples showing that this condition need not hold and give a negative answer to a question of