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The dihedral group of order 8 requires two generators, as represented by this cycle diagram.. In algebra, a finitely generated group is a group G that has some finite generating set S so that every element of G can be written as the combination (under the group operation) of finitely many elements of S and of inverses of such elements.
The fundamental theorem of finitely generated abelian groups can be stated two ways, generalizing the two forms of the fundamental theorem of finite abelian groups.The theorem, in both forms, in turn generalizes to the structure theorem for finitely generated modules over a principal ideal domain, which in turn admits further generalizations.
However, every subgroup of a finitely generated abelian group is in itself finitely generated. In fact, more can be said: the class of all finitely generated groups is closed under extensions. To see this, take a generating set for the (finitely generated) normal subgroup and quotient. Then the generators for the normal subgroup, together with ...
Every finitely presented group is recursively presented, but there are recursively presented groups that cannot be finitely presented. However a theorem of Graham Higman states that a finitely generated group has a recursive presentation if and only if it can be embedded in a finitely presented group. [ 2 ]
In the mathematical area of group theory, the Grigorchuk group or the first Grigorchuk group is a finitely generated group constructed by Rostislav Grigorchuk that provided the first example of a finitely generated group of intermediate (that is, faster than polynomial but slower than exponential) growth.
In mathematics, finiteness properties of a group are a collection of properties that allow the use of various algebraic and topological tools, for example group cohomology, to study the group. It is mostly of interest for the study of infinite groups. Special cases of groups with finiteness properties are finitely generated and finitely ...
Finitely generated group. If there exists a finite set S such that S = G, then G is said to be finitely generated. If S can be taken to have just one element, G is a cyclic group of finite order, an infinite cyclic group, or possibly a group {e} with just one element. Simple group.
Every finitely generated group with a recursively enumerable presentation and insoluble word problem is a subgroup of a finitely presented group with insoluble word problem [14] The number of relators in a finitely presented group with insoluble word problem may be as low as 14 [15] or even 12. [16] [17]