Abstract. Let G be a finite group. A non-negative integer g is called a genus of G if G acts faithfully on a compact orientable surface S g preserving orientation. The set of all such possible genera g d 2 for a finite group G is called the genus spectrum of G; after re-scaling it is called the reduced genus spectrum for G. The reduced genus spectrum of a given finite group G contains all su‰ciently large numbers. We will describe the reduced genus spectrum of finite p-groups of exponent p, where p is a prime, and also for p-groups of maximal class with order less than or equal to p p . IntroductionLet S g be a compact orientable surface of genus g (where g d 2). Let G be a finite group acting faithfully on S g preserving orientation. Then S g can be given a Riemann surface structure X , and the action of G can be realized as an action as automorphisms of X . It is known (from Hurwitz [6]) that if X is a Riemann surface of genus g d 2 then jAutðX Þj c 84ðg À 1Þ, so that all finite groups G acting on S g have order jGj c 84ðg À 1Þ. Wiman [16] and Harvey [5] gave the bound jGj c 4g þ 2 for cyclic groups, and for p-groups a sharper bound was given in Harvey [5] and Kulkarni [8].If G is a finite group, the set spðGÞ :¼ fg d 2 : G acts faithfully on S g preserving orientationg is called the genus spectrum of G. An invariant N 0 ðGÞ was constructed by Kulkarni [8], with the property that all but finitely many g in S ¼ fn : n d 2; n 1 1 mod N 0 g constitute spðGÞ. Two other invariants of G are the minimum genus of G namely,

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.

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Let G m,n,k = Zm ⋉ k Zn be the split metacyclic group, where k is a unit modulo n. We derive an upper bound for the diameter of G m,n,k using an arithmetic parameter called the weight, which depends on n, k, and the order of k. As an application, we show how this would determine a bound on the diameter of an arbitrary metacyclic group. t i=1x ai y bi = x a1+...+at y b1k a 2 +...+a t +...+bt−1k a t +bt .

In this article, we prove that the Schur multiplier of a finite 𝑝-group of maximal class of order p n p^{n} ( 4 ≤ n ≤ p + 1 4\leq n\leq p+1 ) is elementary abelian. The case n = p + 1 n=p+1 settles a question raised by Primož Moravec in an earlier article.