Flnite Group Research Articles

REPRESENTATIONS OF THE AUTOMORPHISM GROUP OF A SUPERSINGULAR K3 SURFACE OF ARTIN-INVARIANT 1 OVER ODD CHARACTERISTIC

In this paper, we prove that the image of the represen- tation of the automorphism group of a supersingular K3 surface of Artin-invariant 1 over odd characteristic p on the global two forms is a flnite cyclic group of order p + 1. Using this result, we deduce, for such a K3 surface, there exists an automorphism which cannot be lifted over a fleld of characteristic 0.

ON CONSTRUCTING REPRESENTATIONS OF THE SYMMETRIC GROUPS

Let G be a symmetric group. In this paper we describe a method that for a certain irreducible characterof G it flnds a subgroup H such that the restriction ofon H has a linear con- stituent with multiplicity one. Then using a well known algorithm we can construct a representation of G afiording ´. m ), l1 > ¢¢¢ > lm > 0; when we have ai parts of size li. Since the number of irreducible characters of a group is equal to the number of conjugacy classes, which in the case of Sn is the number of partitions of n, the irreducible characters of Sn are labelled by partitions of n. We denote the irreducible character labelled by the partition ‚ by (‚), so Irr(Sn) = f(‚) j ‚ ' ng. If G is a flnite group andis an irreducible character of G, an e-cient and simple method to construct representations of flnite groups has been presented in (2). This method is applicable whenever G has a subgroup H such thatH has a linear constituent with multiplicity 1. We call such a subgroup H a ´-subgroup. In practice this algorithm is quite fast when H has a small order, but can be very slow for a large H. For using this method to construct representations of G, we need to flnd a ´-subgroup for each irreducible characterof G. If G is a simple group or a covering group of a simple group, then a ´- subgroup for each nontrivial irreducible characterof G of degree < 32 has been found in (1). Also if ‚ is a partition of n and ‚ 0 is the conjugate

On the character ring of a finite group

Let G be a flnite group and let k be a su-ciently large flnite fleld. Let R(G) denote the character ring of G (i.e. the Grothendieck ring of the category of CG-modules). We study the structure and the representations of the commutative algebra k ›Z R(G). Contents Let G be a flnite group. We denote by R(G) the Grothendieck ring of the category of CG- modules (it is usually called the character ring of G). It is a natural question to try to recover properties of G from the knowledge of R(G). It is clear that two flnite groups having the same character table have the same Grothendieck rings and it is a Theorem of Saksonov (S) that the converse also holds. So the problem is reduced to an intensively studied question in character theory: recover properties of the group through properties of its character table. In this paper, we study the k-algebra kR(G) = k ›Z R(G), where k is a splitting fleld for G of positive characteristic p. It is clear that the knowledge of kR(G) is a much weaker information than the knowledge of R(G). The aim of this paper is to gather results on the representation theory of the algebra kR(G): although most of the results are certainy well-known, we have not found any general treatment of these questions. The blocks of kR(G) are local algebras which are parametrized by conjugacy classes of p-regular elements of G. So the simple kR(G)-modules are parametrized by conjugacy classes of p-regular elements of G. Moreover, the dimension of the projective cover of the simple module associated to the conjugacy class of the p-regular element g 2 G is equal to the number of conjugacy classes of p-elements in the centralizer CG(g). We also prove that the radical of kR(G) is the kernel of the decomposition map kR(G) ! k ›Z R(kG), where R(kG) is the Grothendieck ring of the category of kG-modules (i.e. the ring of virtual Brauer characters of G).

Arc-transitive and s-regular Cayley graphs of valency 5 on Abelian groups

Let G be a flnite group, and let 1G 62 S µ G. A Cayley di-graph i = Cay(G;S) of G relative to S is a di-graph with a vertex set G such that, for x;y 2 G, the pair (x;y) is an arc if and only if yx i1 2 S. Further, if S = S i1 := fs i1 js 2 Sg, then i is undirected. i is conected if and only if G = hsi. A Cayley (di)graph i = Cay(G;S) is called normal if the right regular representation of G is a normal subgroup of the automorphism group of i. A graph i is said to be arc-transitive, if Aut(i) is transitive on an arc set. Also, a graph i is s-regular if Aut(i) acts regularly on the set of s-arcs. In this paper, we flrst give a complete classiflcation for arc-transitive Cayley graphs of valency flve on flnite Abelian groups. Moreover, we classify s-regular Cayley graph with valency flve on an abelian group for each s ‚ 1.