View source for Finitely-generated group

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A group $G$ having a finite generating set $M=\{a_1,\dots,a_d\}$. It thus consists of all products $a_{i_1}^{\epsilon_1}\dots a_{i_n}^{\epsilon_n}$, $i_k\in\{1,\dots,d\}$, $\epsilon_k=\pm1$. If $M$ has $d$ elements, then $G$ is said to be a $d$-generator group. Every generating set of a finitely-generated group contains a finite generating set. One-generator groups are said to be cyclic (they are isomorphic to either the additive group $\mathbf Z$ of integers, or the additive groups $\mathbf Z_n$ of residue classes of integers modulo $n$, $n=1,2,\dots$).

The set of isomorphism classes of $2$-generator groups has the cardinality of the continuum. Every countable group can be isomorphically imbedded in a $2$-generator group; the imbedding group can be chosen to be simple and to be generated by an element of order 2 and one of order 3. Every countable $n$-solvable group (cf. [[Solvable group|Solvable group]]) can be imbedded in a $2$-generator $(n+2)$-solvable group. Every subgroup of finite index in a finitely-generated group is finitely generated. A finitely-generated group has only finitely many subgroups of given finite index. A finitely-generated group can be infinite and periodic; in fact, for every natural number $d\geq2$ and every sufficiently large odd number $n$ there exists an infinite $d$-generator group of exponent $n$ (see [[Burnside problem|Burnside problem]]). A finitely-generated group can be isomorphic to a proper quotient group of itself; in this case it is called non-Hopfian (cf. [[Hopf group|Hopf group]]). There exist solvable non-Hopfian finitely-generated groups. A finitely-generated residually-finite group (see [[Residually-finite group|Residually-finite group]]) is Hopfian. Every finitely-generated group of matrices over a field is residually finite. There exist infinite finitely-generated, and even finitely-presented, groups that are simple (cf. [[Finitely-presented group|Finitely-presented group]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M.I. Kargapolov,   J.I. [Yu.I. Merzlyakov] Merzljakov,   "Fundamentals of the theory of groups" , Springer  (1979)  (Translated from Russian)</TD></TR></table>



====Comments====
For references see also [[Finitely-presented group|Finitely-presented group]].

[[Category:Group theory and generalizations]]

{{SourceattributionEOM|Finitely-generated group}}
[[Category:Mathematics]]