The endomorphisms and the automorphisms of a finite abelian p-group are presented in an efficient way. The order of the automorphism group is computed and its structure investigated. Finally the characteristic subgroups are described for p > 2 and the fully invariant subgroups for any p.

We present explicitly in this expository note the automorphism group of the hypercube Qd of dimension d as a permutation group acting on its 2 d nodes. This group (Qd) acts on the node set Vd of Qd and thus has degree 2 . It is expressed as the binary operation called exponentiation which combines the two symmetric groups S2 (of degree and order 2) and Sd (of degree d and order d!). Speci cally,

If G is a finite subgroup of the automorphism group of a projective curve X and D is a divisor on X stabilized by G, then we compute a simplified formula for the trace of the natural representation of G on the Riemann-Roch space L(D), under the assumption that L(D) is “rational”, D is nonspecial, and the characteristic is “good”. We discuss the partial formulas that result if L(D) is not rational.

We study flat flag-transitive c.c∗-geometries. We prove that, apart from one exception related to Sym(6), all these geometries are gluings in the meaning of [6]. They are obtained by gluing two copies of an affine space over GF(2). There are several ways of gluing two copies of the n-dimensional affine space over GF(2). In one way, which deserves to be called the canonical one, we get a geometr...

In this paper we prove the following theorem. Let S be a linear space. Assume that S has an automorphism group G which is line-transitive and point-imprimitive with k < 9. Then S is one of the following:(a) A projective plane of order 4 or 7, (a) One of 2 linear spaces with v = 91 and k = 6, (b) One of 467 linear spaces with v = 729 and k = 8. In all cases the full automorphism group Aut(S) is ...

The power graph of a group is the graph whose vertex set is the group, two elements being adjacent if one is a power of the other. We observe that nonisomorphic finite groups may have isomorphic power graphs, but that finite abelian groups with isomorphic power graphs must be isomorphic. We conjecture that two finite groups with isomorphic power graphs have the same number of elements of each o...

Much less known, however, is that there is a description of Aut(G), the automorphism group of G. The first compete characterization that we are aware of is contained in a paper by Ranum [1] near the turn of the last century. Beyond this, however, there are few other expositions. Our goal is to fill this gap, thereby providing a much needed accessible and modern treatment. Our characterization o...

In this paper we give a classification of the rank two p-local finite groups for odd p. This study requires the analisis of the possible saturated fusion systems in terms of the outer automorphism group ant the proper F -radical subgroups. Also, for each case in the classification, either we give a finite group with the corresponding fusion system or we check that it corresponds to an exotic p-...

Introduction. The inner automorphism of an element g of a group G is denoted by ig and is given by the formula xig = gxg−1 for all x ∈ G. The mapping π : G→Aut(G), taking each g∈G to the inner automorphism ig, is a homomorphism, Ker(π) = Z(G) and Im(π) = Inn(G). It is easy to check that the automorphism group Aut(G) of a group G with trivial center also is a group with trivial center. Therefore...

This note is part of a general programme to classify the automorphism groups of finite linear spaces. There have been a number of contributions to this programme, including two recent surveys [8, 3]. One of the most significant contributions was the classification of flag-transitive linear spaces [2]. Since then, the effort has been to classify the line-transitive examples. These fall naturally...