About: Finitely generated group

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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. By definition, every finite group is finitely generated, since S can be taken to be G itself. Every infinite finitely generated group must be countable but countable groups need not be finitely generated. The additive group of rational numbers Q is an example of a countable group that is not finitely generated.

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  • Eine endlich erzeugte Gruppe ist ein Objekt aus dem mathematischen Teilgebiet der abstrakten Algebra. Es handelt sich um einen Spezialfall einer Gruppe. (de)
  • 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. By definition, every finite group is finitely generated, since S can be taken to be G itself. Every infinite finitely generated group must be countable but countable groups need not be finitely generated. The additive group of rational numbers Q is an example of a countable group that is not finitely generated. (en)
  • 代数学における有限生成群（ゆうげんせいせいぐん、英: finitely generated group）は、適当な有限部分集合 S を生成系とする群 G を言う。すなわち有限生成群 G の任意の元は、S ∪ S−1（有限集合 S とそれに属する元の逆元の集合 S−1 の合併）の有限個の元の積に書ける。 定義により任意の有限群 G は有限生成である（S = G ととればよい）。任意の有限生成無限群は可算でなければならないが、任意の可算群は必ずしも有限生成でない。実際、有理数全体の成す加法群 Q は有限生成でない可算群の例を与える。 有限生成群の任意の剰余群はまた有限生成である。有限生成群の部分群は有限生成とは限らない。 (ja)
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  • Eine endlich erzeugte Gruppe ist ein Objekt aus dem mathematischen Teilgebiet der abstrakten Algebra. Es handelt sich um einen Spezialfall einer Gruppe. (de)
  • 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. By definition, every finite group is finitely generated, since S can be taken to be G itself. Every infinite finitely generated group must be countable but countable groups need not be finitely generated. The additive group of rational numbers Q is an example of a countable group that is not finitely generated. (en)
  • 代数学における有限生成群（ゆうげんせいせいぐん、英: finitely generated group）は、適当な有限部分集合 S を生成系とする群 G を言う。すなわち有限生成群 G の任意の元は、S ∪ S−1（有限集合 S とそれに属する元の逆元の集合 S−1 の合併）の有限個の元の積に書ける。 定義により任意の有限群 G は有限生成である（S = G ととればよい）。任意の有限生成無限群は可算でなければならないが、任意の可算群は必ずしも有限生成でない。実際、有理数全体の成す加法群 Q は有限生成でない可算群の例を与える。 有限生成群の任意の剰余群はまた有限生成である。有限生成群の部分群は有限生成とは限らない。 (ja)
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  • Endlich erzeugte Gruppe (de)
  • Finitely generated group (en)
  • 有限生成群 (ja)
  • Конечнопорождённая группа (ru)
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