Jay Zimmerman

Groups of small symmetric genus

Group actions on compact surfaces have received considerable attention during the past century. The surface has often carried an analytic structure and been considered a Riemann surface or, equivalently, a complex algebraic curve.

The Groups of Symmetric Genus σ ≤ 8

Let G be a finite group. The symmetric genus σ (G) is the minimum genus of any compact Riemann surface on which G acts faithfully as a group of automorphisms. Here we classify the groups of symmetric genus σ, for all values of σ such that 4 ≤ σ ≤ 8. In addition, we obtain some general results about the partial presentations that groups acting on surfaces must have. We show that a group with even genus and no “large order” elements in its Sylow 2-subgroup has restrictions on its Sylow 2-subgroup. As a consequence, we show that if G is a 2-group with positive symmetric genus, then σ(G) is odd. The software package MAGMA was employed to help with the calculations, and the MAGMA library of small groups was essential to the classification.

THE SYMMETRIC GENUS OF 2-GROUPS

Let G be a finite group. The symmetric genus σ( G ) is the minimum genus of any Riemann surface on which G acts faithfully. We show that if G is a group of order 2 m that has symmetric genus congruent to 3 (mod 4), then either G has exponent 2 m −3 and a dihedral subgroup of index 4 or else the exponent of G is 2 m −2 . We then prove that there are at most 52 isomorphism types of these 2-groups; this bound is independent of the size of the 2-group G . A consequence of this bound is that almost all positive integers that are the symmetric genus of a 2-group are congruent to 1 (mod 4).

The symmetric genus of metacyclic groups

Abstract Let G be a finite group. The symmetric genus of G is the minimum genus of any Riemann surface on which G acts faithfully. Here we determine a useful lower bound for the symmetric genus of a finite group with a cyclic quotient group. The lower bound is attained for the family of K -metacyclic groups, and we determine the symmetric genus of each nonabelian subgroup of a K -metacyclic group. We also provide some examples of other groups for which the lower bound is attained. We use the standard representation of a finite group as a quotient of a noneuclidean crystallographic (NEC) group by a Fuchsian surface group.

Some properties of FC-groups which occur as automorphism groups

We prove that if G is a group such that Aut G is a countably infinite torsion FC-group, then Aut G contains an infinite locally soluble, normal subgroup and hence a nontrivial abelian normal subgroup. It follows that a countably infinite subdirect product of nontrivial finite groups, of which only finitely many have nontrivial abelian normal subgroups, is not the automorphism group of any group. We are concerned with the question: What classes of torsion groups can occur as the full group of automorphisms Aut G of a group G? Robinson [1] has shown that if Aut G is a Cernikov group (a finite extension of a radicable abelian group with the minimal condition), then Aut G is finite. He has also shown that if Aut G is a nilpotent torsion group, then Aut G has finite exponent. The case where G is a group such that Aut G is a countable torsion FC-group (finite conjugate) was examined in a previous paper [2]. It was shown that if G is a group such that Aut G is a countable torsion FC-group, then Aut G has finite exponent if either (1) Aut G has min-2 or (2) v(Aut G) is finite, where vT(H) is the set of all primes dividing the order of some torsion element of H. In addition, an example of a countable torsion FC-group of infinite exponent which occurred as an automorphism group was given to show that the theorem could not be improved. This example contains a nontrivial abelian normal subgroup. The question arises: Can we find an example which has no nontrivial abelian normal subgroups? We will answer this question in the negative. THEOREM. Let G be a group such that Aut G is a countably infinite periodic FC-group. Then either (a) Aut G contains an infinite abelian normal subgroup N, or (b) Aut G contains an infinite, locally soluble, normal {2, 3}-subgroup of bounded exponent and finite index. In either case, Aut G contains a nontrivial abelian normal subgroup. PROOF. Let Q = G/C -InnG, where C is the center of G, and let T be the torsion subgroup of C. It was proven in [2] that Q and Tp are finite for all primes p. Let q = I and let p be any prime which does not divide 2q. Since Tp is finite, we have C = C1 x Tp. It is well known that since IQI and ITp7 are relatively prime, G splits over Tp. It follows that there exists a group G1 containing C1 such that Received by the editors November 26, 1984 and, in revised form, January 20, 1985. 1980 Mathematics Subject Classification. Primary 20F28, 20E26.

THE 2-GROUPS OF ODD STRONG SYMMETRIC GENUS

Let G be a finite group. The strong symmetric genusσ0(G) is the minimum genus of any Riemann surface on which G acts preserving orientation. We show that a 2-group G has strong symmetric genus congruent to 3 (mod 4) if and only if G is in one of 14 families of groups. A consequence of this classification is that almost all positive integers that are the genus of a 2-group are congruent to 1 (mod 4).

The Symmetric Genus of p-Groups

Let G be a finite group. The symmetric genus σ(G) is the minimum genus of any Riemann surface on which G acts. We show that a non-cyclic p-group G has symmetric genus not congruent to 1(mod p 3) if and only if G is in one of 10 families of groups. The genus formula for each of these 10 families of groups is determined. A consequence of this classification is that almost all positive integers that are the genus of a p-group are congruent to 1(mod p 3). Finally, the integers that occur as the symmetric genus of a p-group with Frattini-class 2 have density zero in the positive integers.

Subdirect products of

A compact bordered Klein surface X of genus g ≥ 2 has at most 12(g − 1) automorphisms. A bordered surface for which the bound is attained is said to have maximal symmetry and its full automorphism group is called an M*-group. For M*-groups G and H, we construct a subdirect product L of G and H that is an M*-group. We show that there is a normal subgroup of G whose index is the same as the index of L in the direct product G × H. This general result is specialized to give results about the index of the subdirect product L in the direct product G × H for M*-groups G and H. Then we give a number of sufficient conditions for L to equal G × H and to conclude that the direct product is an M*-group. For example, let G be an M*-group that acts on a bordered Klein surface X. The elements of G that fix a boundary component of X form a dihedral subgroup of order 2q. The number q is called an action index of G. If G and H have relatively prime action indices and one of them is perfect, then the direct product of G and H is an M*-group.

M^*

ABSTRACT Let G be a finite group. The strong symmetric genus σ0(G) is the minimum genus of any Riemann surface on which G acts faithfully and preserving orientation. Let p a prime, and let Jp be the set of integers g for which there is a p-group of strong symmetric genus g. We show that the set Jp has density zero in the set of positive integers.

-groups

Let A and B be finite groups and let S be the set of all extensions of A by B. A group G is called an extension cover of (A, B), if G contains all extensions in S as subgroups of G. A group G is called a minimal extension cover if G is an extension cover of minimal order. Let be the prime factorization of the odd number n and define . The group D n 1 ×…×D n k × Z 2 is the unique minimal extension cover of (Z n , Z 2). This article also constructs a minimal extension cover of (Z 2 n , Z 2). Some conjectures about minimal extension covers are examined as well.