On the automorphism group of inversive planes of odd order

The structure of the fix bundle free automorphism groups of inversive planes of odd order is determined. As a special case of our main result, the automorphism groups with a transitive action on the points of an inversive plane of odd order are essentially determined, and the plane is shown to be miquelian when these have no non-trivial normal subgroups of odd order.     

Finite groups with a splitting automorphism of odd order

In this paper, we prove that a finite group with a splitting automorphism of odd order is solvable. By using this result, we prove that a locally finite group with a splitting automorphism of odd order is locally solvable.     

换位子群为p阶群的有限p-群的自同构群

设G是换位子群为p阶群的有限p-群,确定了AutG的结构,证明了(i)AutG/AutGG≌Zp-1,其中AutGG={α∈AutG|α平凡地作用在G上}.(ii)AutGG/Op(AutG)≌iGL(ni,p)×jSp(2mj,p),其中Op(AutG)是AutG的最大正规p-子群,ni和mj由G惟一确定.     

Symmetric (41,16,6)-designs with a nontrivial automorphism of odd order

If a symmetric (41,16,6)-design has an automorphism σ of odd prime order q then q = 3 or 5. In the case q = 5 we determine all such designs and find a total of 419 nonisomorphic ones, of which 15 are self-dual. When q = 3 a combinatorial explosion occurs and the complete classification becomes impracticable. However, we give a characterization in the particular case when σ has order 3 and fixes 11 points, and find that there are 3,076 nonisomorphic designs with this property, all of them being non self-dual. The other remaining possibility is that σ, of order 3, fixes 5 points. In this case there are 960 orbit matrices (up to isomorphism and duality) and all but one of them yield designs. Here an incomplete investigation shows that in total there are at least 112,000 nonisomorphic designs. © 1993 John Wiley & Sons, Inc.     

finite groups with schmidt group as automorphism group

This paper continues the work of D.MacHale,D.Flannery(Proc.R.Ir.Acad.81A,209—215;83A,189—196)and the author(Proc.R.Ir.Acad,90A,57—62;J.Southwest China Normal University 15,No.1,21.—28)on the topic on“Finite groupswith given Automorphism group”.The following result is proved:Let G be a finite group with Aut G a Schmidt group.Then G is isomorphic toS_3 or Klain 4-group.,or D such that Aut D=Inn D.D is aSchmidt group of order 2~(?)p.S_2(∈Syl_2D)is a normal and special group exoept asupersperspecial group without commutative generators.     

Quasifields associated with an automorphism of the additive group of a finite field

We propose a construction of finite quasifields based on using of special automorphisms of the additive groups of finite fields.     

Exponent of a finite group with a four-group of automorphisms

Let A be a group isomorphic with either S ₄, the symmetric group on four symbols, or D ₈, the dihedral group of order 8. Let V be a normal four-subgroup of A and ?? an involution in ${A\setminus V}$ . Suppose that A acts on a finite group G in such a manner that C _G (V)?=?1 and C _G (??) has exponent e. We show that if ${A\cong S_4}$ then the exponent of G is e-bounded and if ${A\cong D_8}$ then the exponent of the derived group G?? is e-bounded. This work was motivated by recent results on the exponent of a finite group admitting an action by a Frobenius group of automorphisms.     

On Hadamard matrices of order 2(p+1) with an automorphism of odd prime order p

In this paper, we investigate Hadamard matrices of order 2(p + 1) with an automorphism of odd prime order p. In particular, the classification of such Hadamard matrices for the cases p = 19 and 23 is given. Self‐dual codes related to such Hadamard matrices are also investigated. © 2003 Wiley Periodicals, Inc. J Combin Designs 11: 367–380, 2003; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.10052     

Exponent of a finite group with a fixed-point-free four-group of automorphisms

Let e be a positive integer and G a finite group acted on by the four-group V in such a manner that C _G (V) = 1. Suppose that V contains an element v such that the centralizer C _G (v) has exponent e. Then the exponent of G″, the second derived group of G, is bounded in terms of e only.