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| A group $G$ is \emph{finitely generated} if there is a finite subset $X\subseteq G$ such that $X$ generates $G$. That is, every element of $G$ is a product of elements of $X$ and inverses of elements of $X$. Or, equivalently, no proper subgroup of $G$ contains $X$. |
A group $G$ is \emph{finitely generated} if there is a finite subset $X\subseteq G$ such that $X$ generates $G$. That is, every element of $G$ is a product of elements of $X$ and inverses of elements of $X$. Or, equivalently, no proper subgroup of $G$ contains $X$. |
| The set $X$ is then called a \emph{generating set} for $G$. |
The set $X$ is then called a \emph{generating set} for $G$. |
| If $X$ contains inverses of all its elements, then it is said to be \emph{closed under inverses}. |
If $X$ contains inverses of all its elements, then it is said to be \emph{closed under inverses}. |
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| Every finite group is finitely generated, as we can take $X=G$. |
Every finite group is finitely generated, as we can take $X=G$. |
| Every finitely generated group is countable. |
Every finitely generated group is countable. |
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| Any \PMlinkname{quotient}{QuotientGroup} of a finitely generated group is finitely generated. |
Any \PMlinkname{quotient}{QuotientGroup} of a finitely generated group is finitely generated. |
| However, a finitely generated group may have subgroups that are not finitely generated. |
However, a finitely generated group may have subgroups that are not finitely generated. |
| (For example, the free group of rank $2$ is generated by just two elements, |
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| but its commutator subgroup is not finitely generated.) |
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| Nonetheless, a subgroup of finite index in a finitely generated group is necessarily finitely generated; a bound on the number of generators required for the subgroup is given by the |
Nonetheless, a subgroup of finite index in a finitely generated group is necessarily finitely generated; a bound on the number of generators required for the subgroup is given by the |
| \PMlinkname{Schreier index formula}{ScheierIndexFormula}. |
\PMlinkname{Schreier index formula}{ScheierIndexFormula}. |
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| The finitely generated groups all of whose subgroups are also finitely generated are precisely the groups satisfying the maximal condition. This includes all finitely generated nilpotent groups and, more generally, all polycyclic groups. |
The finitely generated groups all of whose subgroups are also finitely generated are precisely the groups satisfying the maximal condition. This includes all finitely generated nilpotent groups and, more generally, all polycyclic groups. |
