Abstract Building upon previous results, a classification is given of finite ‐groups whose elements of order are all fused. This particularly confirms a conjecture of Gross proposed in 1976 on …
Abstract Building upon previous results, a classification is given of finite ‐groups whose elements of order are all fused. This particularly confirms a conjecture of Gross proposed in 1976 on 2‐automorphic 2‐groups, which are 2‐groups with involutions forming a single fusion class. As a consequence, two open problems regarding AT‐groups and FIF‐groups are solved.
We introduce the concept of {\it pseudocover}, which is a counterpart of {\it cover}, for symmetric graphs. The only known example of pseudocovers of symmetric graphs so far was given …
We introduce the concept of {\it pseudocover}, which is a counterpart of {\it cover}, for symmetric graphs. The only known example of pseudocovers of symmetric graphs so far was given by Praeger, Zhou and the first-named author a decade ago, which seems technical and hard to extend to obtain more examples. In this paper, we present a criterion for a symmetric extender of a symmetric graph to be a pseudocover, and then apply it to produce various examples of pseudocovers, including (1) with a single exception, each Praeger-Xu's graph is a pseudocover of a wreath graph; (2) each connected tetravalent symmetric graph with vertex stabilizer of size divisible by $32$ has connected pseudocovers.
The classification of the finite primitive permutation groups of rank $3$ was completed in the 1980s and this landmark achievement has found a wide range of applications. In the general …
The classification of the finite primitive permutation groups of rank $3$ was completed in the 1980s and this landmark achievement has found a wide range of applications. In the general transitive setting, a classical result of Higman shows that every finite imprimitive rank $3$ permutation group $G$ has a unique non-trivial block system $\mathcal{B}$ and this provides a natural way to partition the analysis of these groups. Indeed, the induced permutation group $G^{\mathcal{B}}$ is $2$-transitive and one can also show that the action induced on each block in $\mathcal{B}$ is also $2$-transitive (and so both induced groups are either affine or almost simple). In this paper, we make progress towards a classification of the rank $3$ imprimitive groups by studying the case where the induced action of $G$ on a block in $\mathcal{B}$ is of affine type. Our main theorem divides these rank $3$ groups into four classes, which are defined in terms of the kernel of the action of $G$ on $\mathcal{B}$. In particular, we completely determine the rank $3$ semiprimitive groups for which $G^{\mathcal{B}}$ is almost simple, extending recent work of Baykalov, Devillers and Praeger. We also prove that if $G$ is rank $3$ semiprimitive and $G^{\mathcal{B}}$ is affine, then $G$ has a regular normal subgroup which is a special $p$-group for some prime $p$.
Building upon previous results, a classification is given of finite $p$-groups of which subgroups of order $p$ are all fused. This completes the classification problem dated back to Higman 1963 …
Building upon previous results, a classification is given of finite $p$-groups of which subgroups of order $p$ are all fused. This completes the classification problem dated back to Higman 1963 on the so-called Suzuki $2$-groups, and confirms a conjecture of Gross proposed in 1974. As a consequence, two open problems on AT-groups and FIF-groups are solved.
A complete classification is given of finite groups whose elements are partitioned into three orbits by the automorphism groups, solving the long-standing classification problem initiated by G.\,Higman in 1963. As …
A complete classification is given of finite groups whose elements are partitioned into three orbits by the automorphism groups, solving the long-standing classification problem initiated by G.\,Higman in 1963. As a consequence, a classification is obtained for finite permutation groups of rank $3$ which are holomorphs of groups.
A coset geometry representation of regular dessins is established, and employed to describe quotients and coverings of regular dessins and surfaces. A characterization is then given of face-quasiprimitive regular dessins …
A coset geometry representation of regular dessins is established, and employed to describe quotients and coverings of regular dessins and surfaces. A characterization is then given of face-quasiprimitive regular dessins as coverings of unicellular regular dessins. It shows that there are exactly three O'Nan-Scott-Praeger types of face-quasiprimitive regular dessins which are smooth coverings of unicellular regular dessins, leading to new constructions of interesting families of regular dessins. Finally, a problem of determining smooth Schur covering of simple groups is initiated by studying coverings between $\SL(2,p)$ and $\PSL(2,p)$, giving rise to interesting regular dessins like Fibonacci coverings.
A transitive permutation group is said to be semiprimitive if each of its normal subgroups is either semiregular or transitive.The class of semiprimitive groups properly contains primitive groups, quasiprimitive groups …
A transitive permutation group is said to be semiprimitive if each of its normal subgroups is either semiregular or transitive.The class of semiprimitive groups properly contains primitive groups, quasiprimitive groups and innately transitive groups.The latter three classes of groups of rank $3$ have been classified, forming significant progresses on the long-standing problem of classifying permutation groups of rank $3$.In this paper, a complete classification is given of finite semiprimitive groups of rank $3$ that are not innately transitive, examples of which are certain Schur coverings of certain almost simple $2$-transitive groups, and three exceptional small groups.
This is one of a series of papers which aims towards a classification of imprimitive affine groups of rank $3$. In this paper, a complete classification is given of such …
This is one of a series of papers which aims towards a classification of imprimitive affine groups of rank $3$. In this paper, a complete classification is given of such groups of characteristic $p$ such that the point stabilizer is not $p$-local, which shows that such groups are very rare, namely, the two non-isomorphic groups of the form $2^4{:}\mathrm{GL}_3(2)$ with a unique minimal normal subgroup are the only examples.
A transitive permutation group is said to be semiprimitive if each of its normal subgroups is either semiregular or transitive.The class of semiprimitive groups properly contains primitive groups, quasiprimitive groups …
A transitive permutation group is said to be semiprimitive if each of its normal subgroups is either semiregular or transitive.The class of semiprimitive groups properly contains primitive groups, quasiprimitive groups and innately transitive groups.The latter three classes of groups of rank $3$ have been classified, forming significant progresses on the long-standing problem of classifying permutation groups of rank $3$.In this paper, a complete classification is given of finite semiprimitive groups of rank $3$ that are not innately transitive, examples of which are certain Schur coverings of certain almost simple $2$-transitive groups, and three exceptional small groups.
This is one of a series of papers which aims towards a classification of imprimitive affine groups of rank $3$. In this paper, a complete classification is given of such …
This is one of a series of papers which aims towards a classification of imprimitive affine groups of rank $3$. In this paper, a complete classification is given of such groups of characteristic $p$ such that the point stabilizer is not $p$-local, which shows that such groups are very rare, namely, the two non-isomorphic groups of the form $2^4{:}\mathrm{GL}_3(2)$ with a unique minimal normal subgroup are the only examples.
Abstract Building upon previous results, a classification is given of finite ‐groups whose elements of order are all fused. This particularly confirms a conjecture of Gross proposed in 1976 on …
Abstract Building upon previous results, a classification is given of finite ‐groups whose elements of order are all fused. This particularly confirms a conjecture of Gross proposed in 1976 on 2‐automorphic 2‐groups, which are 2‐groups with involutions forming a single fusion class. As a consequence, two open problems regarding AT‐groups and FIF‐groups are solved.
A coset geometry representation of regular dessins is established, and employed to describe quotients and coverings of regular dessins and surfaces. A characterization is then given of face-quasiprimitive regular dessins …
A coset geometry representation of regular dessins is established, and employed to describe quotients and coverings of regular dessins and surfaces. A characterization is then given of face-quasiprimitive regular dessins as coverings of unicellular regular dessins. It shows that there are exactly three O'Nan-Scott-Praeger types of face-quasiprimitive regular dessins which are smooth coverings of unicellular regular dessins, leading to new constructions of interesting families of regular dessins. Finally, a problem of determining smooth Schur covering of simple groups is initiated by studying coverings between $\SL(2,p)$ and $\PSL(2,p)$, giving rise to interesting regular dessins like Fibonacci coverings.
A complete classification is given of finite groups whose elements are partitioned into three orbits by the automorphism groups, solving the long-standing classification problem initiated by G.\,Higman in 1963. As …
A complete classification is given of finite groups whose elements are partitioned into three orbits by the automorphism groups, solving the long-standing classification problem initiated by G.\,Higman in 1963. As a consequence, a classification is obtained for finite permutation groups of rank $3$ which are holomorphs of groups.
The classification of the finite primitive permutation groups of rank $3$ was completed in the 1980s and this landmark achievement has found a wide range of applications. In the general …
The classification of the finite primitive permutation groups of rank $3$ was completed in the 1980s and this landmark achievement has found a wide range of applications. In the general transitive setting, a classical result of Higman shows that every finite imprimitive rank $3$ permutation group $G$ has a unique non-trivial block system $\mathcal{B}$ and this provides a natural way to partition the analysis of these groups. Indeed, the induced permutation group $G^{\mathcal{B}}$ is $2$-transitive and one can also show that the action induced on each block in $\mathcal{B}$ is also $2$-transitive (and so both induced groups are either affine or almost simple). In this paper, we make progress towards a classification of the rank $3$ imprimitive groups by studying the case where the induced action of $G$ on a block in $\mathcal{B}$ is of affine type. Our main theorem divides these rank $3$ groups into four classes, which are defined in terms of the kernel of the action of $G$ on $\mathcal{B}$. In particular, we completely determine the rank $3$ semiprimitive groups for which $G^{\mathcal{B}}$ is almost simple, extending recent work of Baykalov, Devillers and Praeger. We also prove that if $G$ is rank $3$ semiprimitive and $G^{\mathcal{B}}$ is affine, then $G$ has a regular normal subgroup which is a special $p$-group for some prime $p$.
Building upon previous results, a classification is given of finite $p$-groups of which subgroups of order $p$ are all fused. This completes the classification problem dated back to Higman 1963 …
Building upon previous results, a classification is given of finite $p$-groups of which subgroups of order $p$ are all fused. This completes the classification problem dated back to Higman 1963 on the so-called Suzuki $2$-groups, and confirms a conjecture of Gross proposed in 1974. As a consequence, two open problems on AT-groups and FIF-groups are solved.
We introduce the concept of {\it pseudocover}, which is a counterpart of {\it cover}, for symmetric graphs. The only known example of pseudocovers of symmetric graphs so far was given …
We introduce the concept of {\it pseudocover}, which is a counterpart of {\it cover}, for symmetric graphs. The only known example of pseudocovers of symmetric graphs so far was given by Praeger, Zhou and the first-named author a decade ago, which seems technical and hard to extend to obtain more examples. In this paper, we present a criterion for a symmetric extender of a symmetric graph to be a pseudocover, and then apply it to produce various examples of pseudocovers, including (1) with a single exception, each Praeger-Xu's graph is a pseudocover of a wreath graph; (2) each connected tetravalent symmetric graph with vertex stabilizer of size divisible by $32$ has connected pseudocovers.
Throughout this paper, all algebras considered are non-associative algebras, that is, not necessarily associative algebras.We say that an algebra is auto- morphic if it admits a group of automorphisms which …
Throughout this paper, all algebras considered are non-associative algebras, that is, not necessarily associative algebras.We say that an algebra is auto- morphic if it admits a group of automorphisms which acts transitively on its one-dimensional subspaces.We say that an algebra is finite if it contains finitely many elements.DEFINITION.An algebra is called a quasi division algebra if and only if the non-zero elements of the algebra form a multiplicative quasi-group.The obiect of this paper is to show the following: TEOREM 1.Let A be a finite automorphic algebra with ground field F. If F contains more than two elements, then either A 0 or A is a quasi division algebra.The principal application of this theorem concerns Boen's problem, and its generalizations.A finite p-group P is said to be p-automorphic if it admits a group of automorphisms G which transitively permutes the elements of order p in P. In [1], Boen considered the problem of showing that p-automorphic p-groups of odd order are abelian.Despite the efforts of several workers [1],[2], [13], [16], [17], [6], [15] this problem has remained open up to the present time.A more natural setting of this problem is in terms of algebras.It was shown in [2] that if P is a p-automorphic p-group minimal with respect to being non- abelian, then there is associated with P, an algebra A over the field of p ele- ments with the property that A is anticommutative and A2 0. Moreover if G is the group of automorphisms which acts transitively on the elements of order p in P, then G also acts as a group of operators on the algebra A, in such manner that A and fll (Z (P)) are isomorphic as Z G-modules.Accordingly, Kostrikin introduced the notion of homogeneous algebra, i.e. a finite-dimensional algebra which admits a group of automorphisms transitively permuting its non-zero elements.A proof that finite homogeneous algebras over fields of odd characteristic are zero-algebras would then solve the corresponding problem on p-automorphic p-groups.In [16] Boen's problem was generalized slightly in another direction by considering semi-p-automorphic p-groups (spagroups), i.e. p-groups whose cyclic subgroups of order p are transitively per- muted by a group of automorphisms.The construction of an algebra canoni-
Our purpose here is to study the irreducible representations of semisimple algebraic groups of characteristic p 0, in particular the rational representations, and to determine all of the representations of …
Our purpose here is to study the irreducible representations of semisimple algebraic groups of characteristic p 0, in particular the rational representations, and to determine all of the representations of corresponding finite simple groups. (Each algebraic group is assumed to be defined over a universal field which is algebraically closed and of infinite degree of transcendence over the prime field, and all of its representations are assumed to take place on vector spaces over this field.)
A permutation group is said to be quasiprimitive if each of its nontrivial normal subgroups is transitive. A structure theorem for finite quasiprimitive permutation groups is proved, along the lines …
A permutation group is said to be quasiprimitive if each of its nontrivial normal subgroups is transitive. A structure theorem for finite quasiprimitive permutation groups is proved, along the lines of the O'Nan-Scott Theorem for finite primitive permutation groups. It is shown that every finite, non-bipartite, 2-arc transitive graph is a cover of a quasiprimitive 2-arc transitive graph. The structure theorem for quasiprimitive groups is used to investigate the structure of quasiprimitive 2-arc transitive graphs, and a new construction is given for a family of such graphs.
Abstract We give a self-contained proof of the O'Nan-Scott Theorem for finite primitive permutation groups.
Abstract We give a self-contained proof of the O'Nan-Scott Theorem for finite primitive permutation groups.
Elements a, b of a group G are said to be fused or inverse-fused if there exists σεAut(G) such that a = bσ or a = (b-1)σ respectively. This paper …
Elements a, b of a group G are said to be fused or inverse-fused if there exists σεAut(G) such that a = bσ or a = (b-1)σ respectively. This paper gives a classification of all finite groups in which any two elements of the same order are fused orinverse-fused.
GRAHAM HIGMAN1. Introduction In this paper we shall determine all groups G of order a power of 2 which possess automorphisms that permute their involutions cyclically.The de- termination is complete, …
GRAHAM HIGMAN1. Introduction In this paper we shall determine all groups G of order a power of 2 which possess automorphisms that permute their involutions cyclically.The de- termination is complete, except that we do not exclude the possibility that two or more of the groups that we list may be isomorphic.The investigation is perhaps not without interest simply as an example of the use of linear methods in p-group theory; but the main motivation for it is that some result along these lines is needed by Suzuki in his classification [4] of ZT-groups.It is a pleasure to acknowledge that this paper is, in a direct way, a fruit of the special year in Group Theory organized by the Department of Mathematics at the University of Chicago.A 2-group with only one involution, that is, a eyelie or generalised quaternion group obviously has the property under discussion; and an abelian group has it if and only if it is a direct product of eyelie 2-groups all of the same order.It is convenient to exclude these eases from the beginning, and define a Suzulci 2-group as a non-abelian 2-group with more than one involution, having a eyelie group of automorphisms which permutes its involutions transi- tively.Evidently, the involutions of a Suzuki 2-group G all belong to its center, and so constitute, with the identity, an elementary abelian subgroup fh(G) of order q 2", n > 1.We shall show that fI(G) Z(G) q(G) G', so that G is of exponent 4 and class 2. The automorphism ( which permutes cyclically the q 1 involutions evidently has order divisible by q 1.We shall show that can be taken to have order precisely q 1, and so to be regular.The order of G is either q or qa.In many ways, it would be more satisfactory to impose on G the simpler, weaker condition that the involutions of G are permuted transitively by the full automorphism group of G. Possibly such a relaxation would not bring in any large class of new groups; but the condition seems to be very hard to handle.However, a little of our argument extends to the general ease, and this part has been stated for that ease.The methods used are similar to those involving the associated Lie ring (el.e.g. [2]),but we shall not construct this ring explicitly.The setup, which we shall presuppose, is as follows.If H is a subgroup of the 2-group G, and K a normal subgroup of H with elementary abelian factor group H/K,
Abstract Let G be a finite group and let Aut( G ) be its automorphism group. Then G is called a k -orbit group if G has k orbits (equivalence …
Abstract Let G be a finite group and let Aut( G ) be its automorphism group. Then G is called a k -orbit group if G has k orbits (equivalence classes) under the action of Aut( G ). (For g, h G , we have g ~ h if g a = h for some Aut( G ).) It is shown that if G is a k -orbit group, then k G p + 1, where p is the least prime dividing the order of G . The 3-orbit groups which are not of prime-power order are classified. It is shown that A 5 is the only insoluble 4-orbit group, and a structure theorem is proved about soluble 4-orbit groups.
Let $G$ be a classical group with natural module $V$ over an algebraically closed field of good characteristic. For every unipotent element $u$ of $G$, we describe the Jordan block …
Let $G$ be a classical group with natural module $V$ over an algebraically closed field of good characteristic. For every unipotent element $u$ of $G$, we describe the Jordan block sizes of $u$ on the irreducible $G$-modules which occur as composition factors of $V \otimes V^*$, $\wedge^2(V)$, and $S^2(V)$. Our description is given in terms of the Jordan block sizes of the tensor square, exterior square, and the symmetric square of $u$, for which recursive formulae are known.
Denote by ω(G) the number of orbits of the action of Aut(G) on the finite group G. We prove that if G is a finite nonsolvable group in which ω(G) …
Denote by ω(G) the number of orbits of the action of Aut(G) on the finite group G. We prove that if G is a finite nonsolvable group in which ω(G) ≤5, then G is isomorphic to one of the groups A5, A6, PSL(2, 7), or PSL(2, 8). We also consider the case when ω(G) = 6 and show that, if G is a nonsolvable finite group with ω(G) = 6, then either G ≈ PSL(3, 4) or there exists a characteristic elementary abelian 2-subgroup N of G such that G/N ≈ A5.
Abstract We give a complete classification of the finite 2-groups 𝐺 for which the automorphism group <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>Aut</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>G</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> …
Abstract We give a complete classification of the finite 2-groups 𝐺 for which the automorphism group <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>Aut</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>G</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> \operatorname{Aut}(G) acting naturally on 𝐺 has three orbits. There are two infinite families and one additional group, of order <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mn>2</m:mn> <m:mn>9</m:mn> </m:msup> </m:math> 2^{9} . All of them are Suzuki 2-groups, and they appear (in a different context) in an earlier classification of Dornhoff.
2 2f -1).Here e = 1 if P is isomorphic to the central product of n dihedral groups of order 8 and € = -1 if P is isomorphic to …
2 2f -1).Here e = 1 if P is isomorphic to the central product of n dihedral groups of order 8 and € = -1 if P is isomorphic to the central product ofn-1 dihedral groups of order 8 and a quaternion group.COROLLARY 1.Let p be an odd prime and let P be an extraspecial p-group of exponent p 2 .There is a nonidentity element of PIZ(P) left fixed by every automorphism of P.
Let ω(G) denote the number of orbits on the elements of a group G under the action of its automorphism group. We determine all finite simple groups G such that …
Let ω(G) denote the number of orbits on the elements of a group G under the action of its automorphism group. We determine all finite simple groups G such that ω(G)≤100.
A transitive decomposition is a pair (Γ, 𝒫) where Γ is a graph and 𝒫 is a partition of the arc set of Γ, such that there exists a group …
A transitive decomposition is a pair (Γ, 𝒫) where Γ is a graph and 𝒫 is a partition of the arc set of Γ, such that there exists a group of automorphisms of Γ which leaves 𝒫 invariant and transitively permutes the parts in 𝒫. This paper concerns transitive decompositions where the group is a primitive rank 3 group of ‘grid’ type. The graphs Γ in this case are either products or Cartesian products of complete graphs. We first give some generic constructions for transitive decompositions of products and Cartesian products of copies of an arbitrary graph, and we then prove (except in a small number of cases) that all transitive decompositions with respect to a rank 3 group of grid type can be characterized using these constructions. Furthermore, the main results of this work provide a new proof and insight into the classification of rank 3 partial linear spaces of product action type studied by Devillers.
A partial linear space is a pair (P, L) where P is a non-empty set of points and L is a collection of subsets of P called lines such that …
A partial linear space is a pair (P, L) where P is a non-empty set of points and L is a collection of subsets of P called lines such that any two distinct points are contained in at most one line, and every line contains at least two points.A partial linear space is proper when it is not a linear space or a graph.A group of automorphisms G of a proper partial linear space acts transitively on ordered pairs of distinct collinear points and ordered pairs of distinct non-collinear points precisely when G is transitive of rank 3 on points.In this paper, we classify the finite proper partial linear spaces that admit rank 3 affine primitive automorphism groups, except for certain families of small groups, including subgroups of AΓL1(q).Up to these exceptions, this completes the classification of the finite proper partial linear spaces admitting rank 3 primitive automorphism groups.We also provide a more detailed version of the classification of the rank 3 affine primitive permutation groups, which may be of independent interest.
A partial linear space is a non-empty set of points, provided with a collection of subsets called lines such that any pair of points is contained in at most one …
A partial linear space is a non-empty set of points, provided with a collection of subsets called lines such that any pair of points is contained in at most one line and every line contains at least two points.Graphs and linear spaces are particular cases of partial linear spaces.A partial linear space which is not a graph or a linear space is called proper.In this paper, we give a complete classification of all finite proper partial linear spaces admitting a primitive rank 3 automorphism group of almost simple type.
This is the third volume of a comprehensive and elementary treatment of finite p -group theory. Topics covered in this volume: impact of minimal nonabelian subgroups on the structure of …
This is the third volume of a comprehensive and elementary treatment of finite p -group theory. Topics covered in this volume: impact of minimal nonabelian subgroups on the structure of p -groups, classification of groups all of whose nonnormal subgroups have the same order, degrees of irreducible characters of p -groups associated with finite algebras, groups covered by few proper subgroups, p -groups of element breadth 2 and subgroup breadth 1, exact number of subgroups of given order in a metacyclic p -group, soft subgroups, p -groups with a maximal elementary abelian subgroup of order p 2 , p -groups generated by certain minimal nonabelian subgroups, p -groups in which certain nonabelian subgroups are 2-generator. The book contains many dozens of original exercises (with difficult exercises being solved) and a list of about 900 research problems and themes.
The sets of primitive, quasiprimitive, and innately transitive permutation groups may each be regarded as the building blocks of finite transitive permutation groups, and are analogues of composition factors for …
The sets of primitive, quasiprimitive, and innately transitive permutation groups may each be regarded as the building blocks of finite transitive permutation groups, and are analogues of composition factors for abstract finite groups.This paper extends classifications of finite primitive and quasiprimitive groups of rank at most 3 to a classification for the finite innately transitive groups.The new examples comprise three infinite families and three sporadic examples.A necessary step in this classification was the determination of certain configurations in finite almost simple 2-transitive groups called special pairs.Dedicated to the memory of Jacques Tits
Abstract Building upon previous results, a classification is given of finite ‐groups whose elements of order are all fused. This particularly confirms a conjecture of Gross proposed in 1976 on …
Abstract Building upon previous results, a classification is given of finite ‐groups whose elements of order are all fused. This particularly confirms a conjecture of Gross proposed in 1976 on 2‐automorphic 2‐groups, which are 2‐groups with involutions forming a single fusion class. As a consequence, two open problems regarding AT‐groups and FIF‐groups are solved.
The permutation representations in the title are all determined, and no surprises are found to occur.
The permutation representations in the title are all determined, and no surprises are found to occur.
Abstract 2‐( v,k,1 ) designs admitting a primitive rank 3 automorphism group , where G 0 belongs to the Extraspecial Class, or to the Exceptional Class of Liebeck's Theorem in …
Abstract 2‐( v,k,1 ) designs admitting a primitive rank 3 automorphism group , where G 0 belongs to the Extraspecial Class, or to the Exceptional Class of Liebeck's Theorem in [23], are classified.
This is one of a series of papers which aim toward determining the automorphism groups of Frobenius groups. This paper solves the problem in the case where the Frobenius kernels …
This is one of a series of papers which aim toward determining the automorphism groups of Frobenius groups. This paper solves the problem in the case where the Frobenius kernels are elementary abelian and Frobenius complements are cyclic.