A STRUCTURED DESCRIPTION OF THE GENUS SPECTRUM OF ABELIAN p-GROUPS

Abstract

The genus spectrum of a finite group G is the set of all g such that G acts faithfully on a compact Riemann surface of genus g. It is an open problem to find a general description of the genus spectrum of the groups in interesting classes, such as the abelian p-groups. Motivated by the work of Talu [14] for odd primes p, we develop a general combinatorial machinery, for arbitrary primes, to obtain a structured description of the so-called reduced genus spectrum of abelian p-groups.

Constraining mapping class group homomorphisms using finite subgroups

“…Proof. This follows immediately from Lemma 11 by a result of Müller-Sarkar, who showed that the strong symmetric genus of (Z/3Z) g is 1 + 3 g−1 • µ 0 (g), where µ 0 (g) ≥ 1 when g ≥ 3 [35,Section 9.1].…”

Genus spectra of abelian $p$-groups

“…The genus spectrum has been studied for various classes of groups, such as cyclic p-groups [10], p-groups of cyclic p-deficiency ≤ 2 [12], p-groups of exponent p and p-groups of maximal class [17], split metacyclic groups of order pq with p, q different prime numbers [20]. Genus spectra of abelian p-groups were systematically studied in [15,18].…”

Non-Abelian Simple Groups Act with Almost All Signatures

“…The issue is that computationally, testing the group theoretic condition is much more difficult than the arithmetic conditions, and so it is not currently possible to get the same complete information we have for potential signatures. Indeed, the problem of determining which groups act with which signatures and other related problems is part of a very active area of research, in no small part due to the tremendous advances in computational power over the last few decades, see for example [1], [7], [8], [15].…”