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Étant donnés un système de racines d’une des familles A, B, C, D, F, G et le groupe abélien libre qu’il engendre, on calcule explicitement la série de croissance de ce groupe relativement à Les résultats s’interprètent en termes du polynôme d’Ehrhart de l’enveloppe convexe de .
In this paper we consider duplexes, which are sets with two associative binary operations. Dimonoids in the sense of Loday are examples of duplexes. The set of all permutations carries a structure of a duplex. Our main result asserts that it is a free duplex with an explicitly described set of generators. The proof uses a construction of the free duplex with one generator by planary trees.
Let be a finite group and let be the set of prime divisors of for which . The Gruenberg-Kegel graph of , denoted , is defined as follows: its vertex set is and two different vertices and are adjacent by an edge if and only if contains an element of order . The degree of a vertex in is denoted by and the -tuple is said to be the degree pattern of . Moreover, if is the vertex set of a connected component of , then the largest -number which divides , is said to be an...
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