On the genus spectrum for p-groups of exponent p and p-groups of maximal class

Abstract

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,

Colostrum provision and care of calves among smallholder farmers in the Kaziranga region of Assam, India

“…Unlike previous studies [17,18,19] only a few farmers related early colostrum feeding to neonatal diarrhoea or retained foetal membranes, though the statement "yellow colour (of colostrum) (should be) faded in milk before calf drinks" resonates with the findings of some authors investigating human neonatal health in other areas of India [20]. Investigations in the Kanha region of Madha Pradesh indicate that taking colostrum to make sweets is a normal practice in that region [24], and the practice known to occur throughout the Indian subcontinent [25]. There was no evidence in the study area of the sale of bovine colostrum as a human health food.…”

“…A study in the Kanha region of Madhya Pradesh found that the majority of participants at 37 out of 38 livestock keeper meetings in different villages reported taking quantities of colostrum (referred to as "Khees" or "Cheek" in that region) for home consumption from newly calved cows and buffalo [24]. Typically, the colostrum is boiled with sugar to make confectionery called "ginna" in Hindi, with various other names in other Indian languages [25]. On the Indian Sub-Continent some people ascribe benefits to human health to the consumption of bovine colostrum [26].…”

Reinforcing Microwave Absorption Multiwalled Carbon Nanotube–Epoxy Composites Using Glass Fibers for Multifunctional Applications

“…It is shown that the introduction of six layers of GF into the epoxy resin (thicknesses around 1.5 mm) drastically increased the tensile strength from about 15 MPa to about 400 MPa, leading to a factor of 26.7 increments. Sarkar et al [22] and Mutalikdesai et al [23] also studied the mechanical properties of E-GF-reinforced epoxy composites. They showed that the tensile strength of 60 wt% of GF reinforcement was around 350-400 MPa, which is similar to our results with 55 wt% GF content.…”

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

“…In [10], the problem of determining sp(G) is called the Hurwitz problem associated with G, and the problem of finding the minimum genus min sp(G) of G, also called its strong symmetric genus, has arisen some particular interest. For more details we refer the reader to [1,13], and the references given there.…”

“…We now restrict ourselves to finite p-groups G, where p is a prime. Not too much is known about the genus spectrum of groups within this class, not even if we only look at interesting subclasses, for example those given by bounding a certain invariant such as rank, exponent, nilpotency class, or co-class; see [13].…”