Finitely generated abelian group
Finitely generated abelian group In abstract algebra, an abelian group (G,+) is called finitely generated if there exist finitely many ...
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Cyclic group
Cyclic group In group theory, a cyclic group is a group that can be generated by a single element, in the ...
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Generating set of a group
Generating set of a group In abstract algebra, a generating set of a group $G$ is a subset S ... can be expressed as the product of finitely many elements of S and their inverses ... if S is a subset of a group G, then , the subgroup generated by $S$, is the ...
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Solvable group
Solvable group In the history of mathematics, the origins of group theory lie in the search for a ... etc., and their sums and products). A group is called solvable if it has a ... two definitions are equivalent, since for every group H and every normal subgroup N of ... an equivalent definition is that a solvable group is a group with a composition ...
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Amenable group
Amenable group In mathematics, an amenable group is a topological group G carrying a kind of averaging operation, that is invariant under translations by group elements. In the case where G is not an abelian group, that means translation on a fixed ...
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Grothendieck group
Grothendieck group In mathematics, the Grothendieck group construction in abstract algebra constructs an abelian group from a commutative monoid in the best ... property In its simplest form, the Grothendieck group of a commutative monoid is the universal ... of making that monoid into an abelian group. For example, let M be a ...
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Abelian group
Abelian group In mathematics, an abelian group, also called a commutative group, is a group (G, *) such that a * b = b * a ... notation is usually used. Examples Every cyclic group G is abelian, because if x, ...
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Periodic group
Periodic group In group theory in mathematics, a periodic group is a group in which each element has finite order. The concept of a periodic group should not be confused with that ...
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Hyperbolic group
Hyperbolic group In group theory, a hyperbolic group, also called negatively curved group, word-hyperbolic group, Gromov-hyperbolic group, $\backslash delta$- ...
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Free group
Free group The Cayley graph of the free group on two generators a and b In mathematics, a group G is called free if there is ... only one way as a product of finitely many elements of S and their inverses ... 1 ). Note that the notion of free group is different from the notion free ...
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