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An important step in the proof of the classification of finitely generated abelian groups is that every such torsion-free group is isomorphic to a . A non-finitely generated countable example is given by the additive group of the polynomial ring Z [ X ] {\displaystyle \mathbb {Z} [X]} (the free abelian group of countable rank).
The problem asks, for a particular class of finitely presented groups if there exists an algorithm that, given a finite presentation of a group from the class, computes the rank of that group. The rank problem is one of the harder algorithmic problems studied in group theory and relatively little is known about it. Known results include:
Abelian groups of rank 0 are exactly the periodic abelian groups. The group Q of rational numbers has rank 1. Torsion-free abelian groups of rank 1 are realized as subgroups of Q and there is a satisfactory classification of them up to isomorphism. By contrast, there is no satisfactory classification of torsion-free abelian groups of rank 2. [2]
The rank is a group invariant: it does not depend on the choice of the subgroup. ... In contrast, non-trivial free abelian groups are never divisible, because in a ...
Similarly, the additive group of the integers (, +) is not simple; the set of even integers is a non-trivial proper normal subgroup. [ 1 ] One may use the same kind of reasoning for any abelian group, to deduce that the only simple abelian groups are the cyclic groups of prime order.
class function A class function on a group G is a function that it is constant on the conjugacy classes of G. class number The class number of a group is the number of its conjugacy classes. commutator The commutator of two elements g and h of a group G is the element [g, h] = g −1 h −1 gh. Some authors define the commutator as [g, h] = ghg ...
If a group has nilpotency class at most n, then it is sometimes called a nil-n group. It follows immediately from any of the above forms of the definition of nilpotency, that the trivial group is the unique group of nilpotency class 0, and groups of nilpotency class 1 are exactly the non-trivial abelian groups. [2] [3]
F 4 (q) has a non-trivial graph automorphism when q is a power of 2. These groups are the automorphism groups of 8-dimensional Cayley algebras over finite fields, which gives them 7-dimensional representations. They also act on the corresponding Lie algebras of dimension 14. G 2 (q) has a non-trivial graph automorphism when q is a power of 3