Holomorph

Abstract: 1. Let p be a prime. The finite p-group P is called special if either (i) P is elementary abelian or (ii) the center, commutator subgroup and Frattini subgroup of P all coincide and are elementary abelian. A nonabelian special p-group whose center has order p is called an extraspecial p-group. It is possible to give a uniform treatment of the subject of automorphisms for all the possible isomorphism types of extraspecial p-groups and so some cases that are more or less known are included here. The result when p is odd and P has exponent p leads to an interesting subgroup of the symplectic group Sp (2n, q), q a power of p, n > 1. This subgroup is the semidirect product of Sp (2n — 2, q) and a normal special p-group of order q~ whose center has order q.

Abstract: Let G be a finite group, and $g \geq 2$. We study the locus of genus g curves that admit a G-action of given type, and inclusions between such loci. We use this to study the locus of genus g curves with prescribed automorphism group G. We completely classify these loci for g=3 (including equations for the corresponding curves), and for $g \leq 10$ we classify those loci corresponding to "large" G.