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Nonsolvable Finite Groups all of whose Local Subgroups are Solvable
Abstract
In this paper, the simple N-groups are classified for which e ≧ 3 and 2 ∈ π4. This latter condition means that a Sylow 2-subgroup contains a normal elementary abelian subgroup of order 8 and does not normalize any nonidentity odd order subgroup.
... So let us push the analogy with finite group theory further. The classical N -property was introduced in [Thompson, 1968] where the full classification of the finite, non-soluble N -groups was given, and then proved in a series of subsequent papers: an N -group is a group G all of whose so-called local subgroups are soluble, which in the finite case amounts to requiring that N G (A) be soluble for every abelian subgroup 1 = A ≤ G. The decorations in N • • indicate that we shall focus on connected components, making our condition less restrictive than proper N -ness. ...
... According to the Cherlin-Zilber conjecture, every connected, non-soluble N • • -group should be isomorphic to PSL 2 (K) or SL 2 (K) with K an algebraically closed field. We cannot prove this, and our results will look partial when compared to [Thompson, 1968]. ...
... Another, more restrictive notion of smallness in [Thompson, 1968] was minimal simplicity: a minimal simple group is a simple group all of whose proper subgroups are soluble. The full classification of the finite, minimal simple groups is given in [Thompson, 1968] as a corollary to that of the finite N -groups. ...
We classify a large class of small groups of finite Morley rank: -groups which are the infinite analogues of Thompson's N-groups. More precisely, we constrain the 2-structure of groups of finite Morley rank containing a definable, normal, non-soluble, -subgroup.
... For each odd q, P GL(2, q) has two non-isomorphic stem non-split central extension by a group of order 2. One has semidihedral Sylow 2-subgroups, the other has generalized quaternion and is denoted by SL (2) (2, q). [48]). Let G be a finite group of even order. ...
... (i) G has odd order; (xi) G/O(G) is the group of order 48 named SmallGroup (48,28) in the GAP-database. ...
... (i) G is a generalized quaternion group, (ii) G is isomorphic to the unique perfect group of order 5040 and G/Z(G) ∼ = Alt 7 , (iii) G is isomorphic to the group of order 48 named SmallGroup (48,28) in the GAP-database. ...
Let X be a (projective, geometrically irreducible, nonsingular) algebraic curve of genus defined over an algebraically closed field K of odd characteristic p. Let Aut(X) be the group of all automorphisms of X which fix K element-wise. It is known that if then the p-rank (equivalently, the Hasse-Witt invariant) of X is zero. This raises the problem of determining the (minimum-value) function f(g) such that whenever then X has zero p-rank. For {\em{even}} g we prove that . The {\em{odd}} genus case appears to be much more difficult although, for any genus , if Aut(X) has a solvable subgroup G such that then X has zero p-rank and G fixes a point of X. Our proofs use the Hurwitz genus formula and the Deuring Shafarevich formula together with a few deep results from finite group theory characterizing finite simple groups whose Sylow 2-subgroups have a cyclic subgroup of index 2. We also point out some connections with the Abhyankar conjecture and the Katz-Gabber covers.
... Observe that minimal non-2-solvable simple groups are exactly the minimal simple groups and these groups are classified by Thompson in [39]. ...
... Proof. This is [39,Corollary 1]. □ ...
... By [2], every non-abelian simple group has a minimal simple subgroup, so our starting point here are the minimal simple groups. Thompson [24] determined all minimal simple groups. ...
... Theorem 3.1 [24] The minimal simple groups are as follows: ...
Let G be a finite group. A subgroup H of G is called a TI-subgroup of G if H∩Hx=1 or H for all x∈G. GN=⋂{N⊴G|G/N is nilpotent} is called the nilpotent residual of G. In this paper, we show that a non-abelian finite simple group the nilpotent residuals of all of its subgroups are TI-subgroups is isomorphic to either PSL(2,2p) for some prime p, to PSL(2, 7) or to the Suzuki group Sz(2p) for some odd prime p.
... The proof of the above result uses the fact that a nonabelian finite simple group has even order which is divisible either by 3 or by 5. This is an immediate corollary of the Odd Order Theorem [12] and Thompson's classification of minimal insoluble groups [21]. ...
Given two subgroups H, K of a compact group G, the probability that a random element of H commutes with a random element of K is denoted by . We show that if G is a profinite group containing a Sylow 2-subgroup P, a Sylow 3-subgroup and a Sylow 5-subgroup such that and are both positive, then G is virtually prosoluble (Theorem 1). Furthermore, if G is a prosoluble group in which for every subset there is a Hall -subgroup and a Hall -subgroup such that , then G is virtually pronilpotent (Theorem 2).
... can be answered by examining all 2-generated subgroups of the group [7]. A more complicated and celebrated result is Thompson's classification of N -groups, finite groups all of whose subgroups are either solvable or Fitting-free [16]. Another historically famous line of research involved the question of what can be said about a non-abelian group all of whose proper subgroups are abelian. ...
We define and investigate the property of being “exponent-critical” for a finite group. A finite group is said to be exponent-critical if its exponent is not the least com-
mon multiple of the exponents of its proper non-abelian subgroups. We explore properties of exponent-critical groups and give a characterization of such groups. This characterization generalizes a classical result of Miller and Moreno on minimal non-abelian groups;
interesting families of p-groups appear.
... It is well-known that G is a minimal nonsolvable group if and only if G/Φ(G) is a finite minimal simple group (that is, a non-abelian finite simple group S such that S is minimal nonsolvable), where Φ(G) is the Ftattini subgroup of G that is the intersection of the maximal subgroups of G. In 1968, J. Thompson [27,Corollary 1] proved that S is a minimal simple group if and only if S is isomorphic to a group in the following list T (we will call it as the Thompson list): ...
Let be a set of primes. According to H. Wielandt, a subgroup H of a finite group X is called a -submaximal subgroup if there is a monomorphism into a finite group Y such that is subnormal in Y and for a -maximal subgroup K of Y. In his talk at the well-known conference on finite groups in Santa-Cruz (USA) in 1979, Wielandt posed a series of open questions and among them the following problem: to describe the -submaximal subgroup of the minimal nonsolvable groups and to study properties of such subgroups: the pronormality, the intravariancy, the conjugacy in the automorphism group etc. In the article, for every set of primes, we obtain a description of the -submaximal subgroup in minimal nonsolvable groups and investigate their properties, so we give a solution of Wielandt's problem.
... For our final result, recall that a finite group S is minimal nonabelian simple if it is non-abelian simple and every proper subgroup of S is soluble; such groups were classified by J. G. Thompson [12,13] well before the classification of all finite simple groups. ...
We study infinitely iterated wreath products of finite permutation groups with respect to product actions. In particular, we prove that, for every non-empty class of finite simple groups , there exists a finitely generated hereditarily just infinite profinite group W with composition factors in such that any countably based profinite group with composition factors in can be embedded into W. Additionally we investigate when infinitely iterated wreath products of finite simple groups with respect to product actions are co-Hopfian or non-co-Hopfian.
... The proof of the above result uses the fact that a nonabelian finite simple group has even order which is divisible either by 3 or by 5. This is an immediate corollary of the Odd Order Theorem [13] and Thompson's classification of minimal insoluble groups [22]. ...
Given two subgroups H,K of a compact group G, the probability that a random element of H commutes with a random element of K is denoted by Pr(H,K). We show that if G is a profinite group containing a Sylow 2-subgroup P, a Sylow 3-subgroup and a Sylow 5-subgroup such that and are both positive, then G is virtually prosoluble (Theorem 1.1). Furthermore, if G is a prosoluble group in which for every subset there is a Hall -subgroup and a Hall -subgroup such that , then G is virtually pronilpotent (Theorem 1.2).
... In a series of articles (see [23]), Thompson described all non-Abelian simple groups all of whose local subgroups 3 are soluble. In particular, he determined the minimal simple groups. ...
... In Pakianathan & Shankar (2000), we find criteria based on J. Thompson's deep result (Thompson, 1968) on minimal simple groups, for the positive integer m to be a solvable group order, that is every group of order m is solvable: m is a solvable group order if and only if m is not a multiple of any of (i) 2 p (2 p -1), p a prime (ii) 3 p (3 2p -1)/2, p an odd prime (iii) p(p 2 -1)/2, p a prime >3 and p ≡ 2 or 3 (mod 5) (iv) 2 4 3 3 13 (v) 2 2p (2 2p + 1)(2 2p -1), p an odd prime. The On-Line Encyclopedia of Integer Sequences (OEIS) has a list of many such m (A056866). ...
... can be answered by examining all 2-generated subgroups of the group [7]. A more complicated and celebrated result is Thompson's classification of N -groups, finite groups all of whose subgroups are either solvable or Fitting-free [16]. Another historically famous line of research involved the question of what can be said about a non-abelian group all of whose proper subgroups are abelian. ...
We define and investigate the property of being 'exponent-critical' for a finite group. A finite group is said to be exponent-critical if its exponent is not the least common multiple of the exponents of its proper non-abelian subgroups. Recent computational work about varieties of groups relies on the fact that the groups in question are not exponent-critical. In this paper we explore properties of exponent-critical groups and give a characterization of such groups.
... Recently, the author and others considered the relation between the degrees of proper subgroups and group structure; see [ 8 ] and [ 9 ] for instance. We also consider the impact of the properties of the proper subgroups on the structure of finite group. ...
Finite groups with real-valued irreducible characters of prime degree are classified by Dolfi, Pacifici and Sanus. In this paper, the structures of finite groups whose all proper subgroups have at most two real-valued-irreducible-character degrees are determined.
... Minimal simple groups have been classified in [14] and in this paper, we provide a similar classification for finite unitary rings, analyzing the simplicity of their groups of units and the groups of units of their subrings. More precisely, we characterize the structure of finite rings R such that R * is not a solvable group but S * is a solvable group for every proper unitary subring S of R. A similar classification for rings with minimal non-nilpotent groups of units has been done in [1]. ...
Let R be a finite unitary ring whose group of units is not solvable but all groups of units of all its proper subrings are solvable. In this paper, we classify these rings and show that all finite rings of order pn for n < 5 and some of order p6 are in this class of rings.
... In [3], Baer proved that G is supersoluble if and only if every two-generated subgroup of G is supersoluble. A similar result for the solubility property is provided by John G. Thompson in [15,Corollary 2] which is an outstanding criterion for the solubility of finite groups. The result reads as follows: ...
Let G be a finite group, and let x be an element of G. Denote by SolG(x) the set of all y ∈ G such that the group generated by x and y is soluble. We investigate the influence of SolG(x) on the structure of G.
(Full-text- view-only: https://rdcu.be/dmTQL)
... The minimal simple groups were completely classified in [18]. They are ...
Given a non-empty set X of a group G, a subgroup A is said to be X-permutable (respectively, X-h-permutable) with a subgroup B, if ABx=BxA, for some element x∈X (respectively, x∈X∩⟨A,B⟩). A subgroup A of a group G is said to be X-permutable (respectively, X-h-permutable) in G if A is X-permutable (respectively, X-h-permutable) with every subgroup of G. In this paper, we study the structure of a finite group G with all its Schmidt subgroups G-permutable (respectively, G-h-permutable).
... We recall that a minimal simple group is a finite non-abelian simple group all of whose proper subgroups are soluble. These groups have been classified by Thompson in [26]. is both a non-trivial γ * k -value and a non-trivial δ * k−1 -value. ...
Let G be a profinite group. The coprime commutators and are defined as follows. Every element of G is both a -value and a -value. For , let X be the set of all elements of G that are powers of -values. An element a is a -value if there exist and such that a=[x,g] and . For , let Y be the set of all elements of G that are powers of -values. The element a is a -value if there exist such that a=[x,y] and . In this paper we establish the following results. A profinite group G is finite-by-pronilpotent if and only if there is k such that the set of -values in G has cardinality less than . A profinite group G is finite-by-(prosoluble of Fitting height at most k) if and only if there is k such that the set of -values in G has cardinality less than .
... In this article, we only use the fact that any homomorphic image of a solvable group is solvable and the Kaplan-Levy criterion (2010) (generalising Thompson's (1968, Corollary 3)) according to which a finite group G is unsolvable iff it contains elements a, b, c such that o G (a) = 2, o G (b) is an odd prime, o G (c) > 1 and coprime to both 2 and o G (b), and abc is the identity of G. ...
Our concern is the problem of determining the data complexity of answering an ontology-mediated query (OMQ) formulated in linear temporal logic LTL over (Z,<) and deciding whether it is rewritable to an FO(<)-query, possibly with some extra predicates. First, we observe that, in line with the circuit complexity and FO-definability of regular languages, OMQ answering in AC0, ACC0 and NC1 coincides with FO(<,≡)-rewritability using unary predicates x ≡ 0 (mod n), FO(<,MOD)-rewritability, and FO(RPR)-rewritability using relational primitive recursion, respectively. We prove that, similarly to known PSᴘᴀᴄᴇ-completeness of recognising FO(<)-definability of regular languages, deciding FO(<,≡)- and FO(<,MOD)-definability is also PSᴘᴀᴄᴇ-complete (unless ACC0 = NC1). We then use this result to show that deciding FO(<)-, FO(<,≡)- and FO(<,MOD)-rewritability of LTL OMQs is ExᴘSᴘᴀᴄᴇ-complete, and that these problems become PSᴘᴀᴄᴇ-complete for OMQs with a linear Horn ontology and an atomic query, and also a positive query in the cases of FO(<)- and FO(<,≡)-rewritability. Further, we consider FO(<)-rewritability of OMQs with a binary-clause ontology and identify OMQ classes, for which deciding it is PSᴘᴀᴄᴇ-, Π2p- and coNP-complete.
... Remark 2.2. In [21] Thompson defined a notion of a minimal simple group as a non-abelian finite simple group all of whose proper subgroups are solvable. As we will see, minimal nonsolvable groups do not have to be simple, hence we consider strictly larger class of groups. ...
It has been shown by several authors that there exists a non-solvable Bieberbach group of dimension 15. In this note we show that this is in fact a minimal dimension for such kind of groups.
... The solubility graph Γ S (G) associated with a group G is a simple graph whose vertices are the elements of G, and there is an edge between two distinct elements x and y if and only if the subgroup x, y is soluble. In [15] Thompson proved that a finite group G is soluble if and only if for every x, y ∈ G the subgroup x, y is soluble. This implies that a finite group G is soluble if and only if the graph Γ S (G) is complete. ...
The solubility graph ΓS(G) associated with a finite group G is a simple graph whose vertices are the elements of G, and there is an edge between two distinct vertices if and only if they generate a soluble subgroup. In this paper, we focus on the set of neighbors of a vertex x which we call the solubilizer of x in G, SolG(x), investigating both arithmetic and structural properties of this set.
... [12,13]) vitalized the field of finite group theory in the 1960s and still exert a powerful influence on the studies of group structure. As is well-known, local subgroups also lie at the heart of the classification of finite simple groups, where Thompson's classification of N -groups [14] is one of the most remarkable contributions. ...
Let G be a finite group and S be a proper subgroup of G. A group G is called an S-(CAP)-group if every local subgroup of G is either a CAP-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.
... On the other hand, some authors considered the influence of subgroups with certain properties on the structure of finite groups. In particular, the structures of finite groups are determined if the proper subgroups are all abelian [1] or solvable [27,31] or supersolvable [6] or nilpotent [18] or minimal non-nilpotent [2]. ...
Huppert and Qian et al. classified finite groups for which all irreducible character degrees are consecutive. The aim of this paper is to determine the structure of finite groups whose irreducible character degrees of their proper subgroups are all consecutive.
... In [3], Baer proved that G is supersoluble if and only if every two-generated subgroup of G is supersoluble. A similar result for the solubility property is provided by John G. Thompson in "N-group paper" [13] which is an outstanding criterion for the solubility of finite groups. This result as follows: ...
Let G be a finite group and x be an element of G. Denote by \Sol_G (x) the set of all elements satisfying this property that is a soluble subgroup of G. This paper investigates the structure of finite groups G is influenced by \Sol_G (x).
Every four years leading researchers gather to survey the latest developments in all aspects of group theory. Since 1981, the proceedings of these meetings have provided a regular snapshot of the state of the art in group theory and helped to shape the direction of research in the field. This volume contains selected papers from the 2022 meeting held in Newcastle. It includes substantial survey articles from the invited speakers, namely the mini course presenters Michel Brion, Fanny Kassel and Pham Huu Tiep; and the invited one-hour speakers Bettina Eick, Scott Harper and Simon Smith. It features these alongside contributed survey articles, including some new results, to provide an outstanding resource for graduate students and researchers.
We show that for any finite group G and for any d there exists a word such that a d-tuple in G satisfies w if and only if it generates a solvable subgroup. In particular, if G itself is not solvable, then it cannot be obtained as a quotient of the one relator group . As a corollary, the probability that a word is satisfied in a fixed non-solvable group can be made arbitrarily small, answering a question of Alon Amit.
In this paper, the structure of finite groups in which the order of each non-normal subgroup has at most 3 prime factors is determined.
Lewis determined the structure of finite groups whose irreducible character odd-degrees are square-free. In this paper, we give an identification of the structure of non-solvable groups whose proper subgroups have only square-free degrees.
A subgroup H in a finite group G is S -conditionally permutable if for every there exists a Sylow p -subgroup P in G such that HP=PH . We study the structure of a finite group G whose all Schmidt subgroups are S -conditionally permutable.
For a nontrivial finite group G, the intersection graph of G is the simple undirected graph whose vertices are the nontrivial proper subgroups of G and two vertices are joined by an edge if and only if they have a nontrivial intersection. In a finite simple graph , the clique number of is denoted by . In this paper we show that if G is a finite group with , then G is solvable. As an application, we characterize all non-solvable groups G with . Moreover, we determine all finite groups G with .
In this paper, the structure of a finite group G all of whose minimal subgroups are hereditarily G-permutable is studied.
A finite group G is called an SNS-group if every subgroup of G is either subnormal in G or self-normalizing. In this article, we investigate the structure of finite groups which are not SNS-groups but all of whose maximal subgroups of even order are SNS-groups. In addition, we determine the minimal simple groups all of whose second maximal subgroups (respectively, second maximal subgroups of even order) are SNS-groups.
We give a computer-assisted proof that if G is a finite group of order 8pq, where p and q are distinct primes, then every connected Cayley graph on G has a hamiltonian cycle.
We give some sufficient tests for the solubility and supersolubility of a finite group G
provided that all maximal subgroups of some Sylow subgroups in G are hereditarily
G -permutable or G -permutable in G .
In this paper, the structures of non-solvable groups whose all proper subgroups have at most seven character values are identified.
A finite group G G is called an MSN-group if all maximal subgroups of the Sylow subgroups of G G are subnormal in G G . In this article, we investigate the structure of finite groups G G such that G G is a non-MSN-group of even order in which every maximal subgroup of even order is an MSN-group. In addition, we determine the minimal simple groups all of whose second maximal subgroups are MSN-groups.
Let be a finite group admitting a coprime automorphism . Let denote the set of all commutators , where belongs to an ‐invariant Sylow subgroup of . We show that is soluble or nilpotent if and only if any subgroup generated by a pair of elements of coprime orders from the set is soluble or nilpotent, respectively.
Let G be a finite group and assume that a group of automorphisms A is acting on G such that A and G have coprime orders. We prove that the fact of imposing specific properties on the second maximal A-invariant subgroups of G determines that G is either soluble or isomorphic to a few non-soluble groups such as PSL(2, 5) or SL(2, 5).
Let G be a finite insoluble group with soluble radical R(G). In this paper we investigate the soluble graph of G, which is a natural generalisation of the widely studied commuting graph. Here the vertices are the elements in G∖R(G), with x adjacent to y if they generate a soluble subgroup of G. Our main result states that this graph is always connected and its diameter, denoted δS(G), is at most 5. More precisely, we show that δS(G)⩽3 if G is not almost simple and we obtain stronger bounds for various families of almost simple groups. For example, we will show that δS(Sn)=3 for all n⩾6. We also establish the existence of simple groups with δS(G)⩾4. For instance, we prove that δS(A2p+1)⩾4 for every Sophie Germain prime p⩾5, which demonstrates that our general upper bound of 5 is close to best possible. We conclude by briefly discussing some variations of the soluble graph construction and we present several open problems.
In his address at the International Congress of Mathematicians at Amsterdam [1] Professor R. Brauer proposed a problem of characterizing various groups of even order by the properties of the involutions contained in these groups and he gave characterizations of the general projective linear groups of low dimensions along these lines. The detail of the one-dimensional case has been published in [5], but the two-dimensional case has not appeared yet in detail. His work was followed by Suzuki [7], Feit [6] and Walter [11]. The present paper is a continuation of [73 and discusses a characterization of the two-dimensional projective unitary group over a finite field of characteristic 2. The precise conditions which characterize the group in question will be stated in the first section. The method employed here is similar to the one used in [7]. An application of group characters provides a formula for the order. However a difficulty comes in when one attempts to identify the group. In order to overcome this difficulty we will use a method primarily designed to study a class of doubly transitive permutation groups (cf. [9]). We need also a group theoretical characterization of a class of doubly transitive groups called ( ZT )-groups. This is a generalization of a result in [8], and may be of in dependent interest.
The object of this paper is to put in relief one of the ideas which has been very helpful in studying simple groups, viz. using factorizations of p-solvable groups to obtain information about the subgroups of a simple group which contain a given Sp-subgroup. Since the idea is so simple, it seems to deserve a simple exposition.
Let p be a prime, and let S be a Sylow p -subgroup of a finite group G . J. Thompson ( 13 ; 14 ) has introduced a characteristic subgroup J R (S) and has proved the following results:
(1.1) Suppose that p is odd. Then G has a normal p-complement if and only if C(Z(S)) and N(J R (S)) have normal p-complements .
The list of known finite simple groups other than the cyclic, alternating, and Mathieu groups consists of the classical groups which are (projective) unimodular, orthogonal, symplectic, and unitary groups, the exceptional groups which are the direct analogues of the exceptional Lie groups, and certain twisted types which are constructed with the aid of Lie theory (see §§3 and 4 below). In this article, it is proved that each of these groups is generated by two of its elements. It is possible that one of the generators can be chosen of order 2, as is the case for the projective unimodular group (1), or even that one of the generators can be chosen as an arbitrary element other than the identity, as is the case for the alternating groups. Either of these results, if true, would quite likely require methods much more detailed than those used here.
If the finite group G has a 2-Sylow subgroup S of order 2 a+1 , containing a cyclic subgroup of index 2, then in general S may be one of the following six types [8]:
(i) cyclic; (ii) Abelian of type ( a , 1), a > 1; (iii) dihedral ¹ ; (iv) generalized quaternion; (v) {α, β}, α ² a = β ² , α ² a−1 +1, a ≧ 3;
(vi) {α, β}, α ² a = β ² , α ² a−1 +1, a ≧ 3.
This is the third of three parts. A description of the contents may be found in the introduction given in the first part. The first part was published in this journal, volume 2, pp. 85–151 (1965). The second part was published in this journal, volume 2, pp. 218–270 (1965).
Let G denote a group of order a power of the prime p , and let G ′ be the derived group of G . The lower central series of G will be written
For any subgroup H of G we denote by P ( H ) the subgroup of H generated by all elements x p as x runs through H , and by Φ( H ) the Frattini subgroup of H . We write ( H :Φ( H )) = p d(H) ; thus d ( H ) is the minimal number of generators of H .
1. Fragestellung. RICHARD BRAUER hat ktirzlich*) auf eine Beziehung hingewiesen, die zwischen den Fixpunktzahlen von vertauschbaren involutorischen Automorphismen einer Gruppe ~ yon ungerader Ordnung h besteht : Wenn die Automorphismengruppe von ~ eine Vierergruppe | enth~lt und wenn die in G0 enthaltenen Untergruppen der Ordnung 2 mit @,, @~, (~8 bezeichnet werden und allgemein ], die Anzahl der Fixpunkte von (~ in bedeutet, so ist
Let (; be a group, which we shall write additively, although it is not assumed to be abelian; and suppose G possesses an automorphism ~ which leaves only the neutral element 0 of G fixed. If ~ has order 2 and G is finite, then G is abelian; for G then has odd order, and thus every element of G can be uniquely halved; and it has been shown elsewhere [2] that a group in which every element has a unique half and which admits an involutory automorphism fixing no element other than 0, must be abelian. It is then natural to ask how much can be deduced about G if has order q > 2. We consider here the case q = 3; the method is quite elementary. Mappings of G into G are denoted by Greek letters and written as right-hand factors: t stands for the identity mapping, 0 for the null (or trivial) endomorphism, which maps all G on 0. Multiplication of mappings then has its usual meaning, that is g(~ v) = (g ~) for all elements g ~ G and a|l mappings #, v. We also introduce the addition of m~ppings defined by g(/~ + ~,) -- g # + g ~, . Only mappings of G into itself that map 0 on 0 are considered. They form a nearring, that is an algebraic system with two operations, addition and multiplication; with respect to addition it is a not necessarily abelian group with neutral element 0; with respect to multiplication it is a semigroup, in our case with neutrai element t and zero 0; and one distributive law 1)
In 1903 H. F. Blichfeldt proved the following brilliant theorem : Let G be a matrix group of order g and of degree n. Let p be a prime divisor of g such that Then G contains the abelian normal p -Sylow subgroup. In 1941 applying his modular theory of the group representation, R. Brauer improved this theorem in the case in which p divides g to the first power only. Further in 1943 H. F. Tuan improved this result of R. Brauer one step more.
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SO that 121: 3i 0 | « ƒ>. Let © 0 = 2I 0 £)o/6 = 3ïoS/6XS)o/S. Since £) 0 /S is represented faithfully on Lemma 3.7 of [20] implies that 35o/S is represented faithfully on * By the minimality of SD 0 , we also get £)oC [©o
- = Let
- Ok
- do
- Oe
Let 3ïo = OK(5DO/OE) = Cfc(©o/©i), SO that 121: 3i 0 | « ƒ>. Let © 0 = 2I 0 £)o/6 = 3ïoS/6XS)o/S. Since £) 0 /S is represented faithfully on /?*(©), Lemma 3.7 of [20] implies that 35o/S is represented faithfully on *,(®)nC(8t 0 ) = JB, say. By hypothesis, [31, »] = 1. Let 8 = (S3ï£)o. Then S3t C Cs(33) < 8. By the minimality of SD 0, we also get £)oC [©o, 3t]£C C(SB), so 8C C(JB). This contradiction completes the proof.
TiZ). Thus, | M t \ = 4. Since Afi+Af^ admits ©, we get Jlfi©Jlfir 2 = Af, and so | Af | =16. Since 7\Z centralizes M± and since AfiZi admits 7\Z, it follows that | CM(T\Z)\ =8. Hence, Af is not a free F 2 33i-module
- Thus
- M\ T\z
Thus, T\Z centralizes M\ % so we may view M\ as a i^O-module, where Q = 0/<i4, TiZ). Thus, | M t \ = 4. Since Afi+Af^ admits ©, we get Jlfi©Jlfir 2 = Af, and so | Af | =16. Since 7\Z centralizes M± and since AfiZi admits 7\Z, it follows that | CM(T\Z)\ =8. Hence, Af is not a free F 2 33i-module. Choose rnE.M\—C Ml (Z).
tf (2)«r(8) centralizes $,(2) and inverts both ^(D and $,(3), while ri 2 r 2 3r 3 i inverts $. Thus, (c) and (d) also hold. We may now assume that ty'?* 1. We apply the portion of the proof already
- Also
Also, r tf (1)^(8)r tf (2)«r(8) centralizes $,(2) and inverts both ^(D and $,(3), while ri 2 r 2 3r 3 i inverts $. Thus, (c) and (d) also hold. We may now assume that ty'?* 1. We apply the portion of the proof already completed to $/$'. Let Xi, X 2, X 3 be the characters of X on $/$' defined previously. We may assume that $' is of order p. Then QiOP) is of exponent p. If jfli^)! =£ 1968] NONSOLVABLE FINITE GROUPS 415
©) contains no noncyclic abelian sub-group of order 8. If 0 2 (©) contains no four-subgroup, then 0 2 (©) is necessarily a quaternion group, so © contains no noncyclic abelian subgroup of order 8
- Finally
Finally, suppose that 0 2 (©) contains no noncyclic abelian sub-group of order 8. If 0 2 (©) contains no four-subgroup, then 0 2 (©) is necessarily a quaternion group, so © contains no noncyclic abelian subgroup of order 8. Let 31 be a four-subgroup of
Z(a«))<«. Since Z(ft«)çz( §I tf )
- Oe-C« Let
Let OE-C«(Z(a«))<«. Since Z(ft«)çz( §I tf ), it follows that <£C C(Z(5«)).
The theory of groups
- M Hall
- Jr
M. Hall, Jr., The theory of groups, Macmillan, New York, 1959.
On a conjecture of Frobenius
- . J Rust
J. Rust, On a conjecture of Frobenius, Ph.D. Thesis, University of Chicago, 1966.
Finite groups with nilpotent centralizers 425-470. 35 # 1 Finite groups of even order whose Sylow 2-subgroups are independent
- M Suzuki
M. Suzuki, Finite groups with nilpotent centralizers, Trans. Amer. Math. Soc. 99(1961), 425-470. 35 # 1 Finite groups of even order whose Sylow 2-subgroups are independent, Ann. of Math (80) 1 (1964), 56-77.
On a conjecture arising from a theorem of Frobenius
- . R Zemlin
R. Zemlin, On a conjecture arising from a theorem of Frobenius, Ph.D. Thesis, Ohio State University, 1954 (unpublished).
Some applications of the theory of blocks of characters of finite groups. II
- Brauer, Richard