Proficient presentations and direct products of finite groups
نویسندگان
چکیده
منابع مشابه
Proficient Presentations and Direct Products of Finite Groups
Let G be a finite group, F a free group of finite rank, R the kernel of a homomorphism tp of F onto G, and let [ i i , f] , [R,R] denote mutual commutator subgroups. Conjugation in F yields a G-module structure on R/[R,R]; let da[R/[R, R]) be the number of elements required to generate this module. Define d(R/[R, F]) similarly. By an earlier result of the first author, for a fixed G, the differ...
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Complete presentations provide a natural solution to the word problem in monoids and groups. Here we give a simple way to construct complete presentations for the direct product of groups, when such presentations are available for the factors. Actually, the construction we are referring to is just the classical construction for direct products of groups, which has been known for a long time, bu...
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The direct product of a free group and a polycyclic group is known to be coherent. This paper shows that every finitely generated subsemigroup of the direct product of a virtually free group and an abelian group admits a finite Malcev presentation. (A Malcev presentation is a presentation of a special type for a semigroup that embeds into a group. A group is virtually free if it contains a free...
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Although (G, dS) is not a geodesic metric space, its Cayley graph ΓS(G) is, and it is the geodesics that provide the natural relationship between dS and the Cayley graph. Let G be the vertex set on the Cayley graph. Two vertices g and h are adjacent in ΓS(G) precisely when g −1h ∈ S or h−1g ∈ S, in which case dS(g, h) = 1. Inductively, it follows that dS(g, h) = n precisely when the shortest pa...
متن کامل16A. Direct products and Classification of Finite Abelian Groups
Definition. Let G and H be groups. Their direct product is the group G×H defined as follows. As a set G×H = {(g, h) : g ∈ G, h ∈ H} is just the usual Cartesian product of G and H (the set of ordered pairs where the first component lies in G and the second component lies in H). The group operation on G×H is defined by the formula (g1, h1)(g2, h2) = (g1g2, h1h2) for all g1, g2 ∈ G and h1, h2 ∈ H....
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ژورنال
عنوان ژورنال: Bulletin of the Australian Mathematical Society
سال: 1999
ISSN: 0004-9727,1755-1633
DOI: 10.1017/s0004972700036315