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A group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040350/f0403501.png" /> having a finite generating set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040350/f0403502.png" />. It thus consists of all products <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040350/f0403503.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040350/f0403504.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040350/f0403505.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040350/f0403506.png" /> has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040350/f0403507.png" /> elements, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040350/f0403508.png" /> is said to be a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040350/f04035010.png" />-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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040350/f04035011.png" /> of integers, or the additive groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040350/f04035012.png" /> of residue classes of integers modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040350/f04035013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040350/f04035014.png" />).
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040350/f04035015.png" />-generator groups has the cardinality of the continuum. Every countable group can be isomorphically imbedded in a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040350/f04035016.png" />-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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040350/f04035018.png" />-solvable group (cf. [[Solvable group|Solvable group]]) can be imbedded in a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040350/f04035019.png" />-generator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040350/f04035020.png" />-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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040350/f04035021.png" /> and every sufficiently large odd number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040350/f04035022.png" /> there exists an infinite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040350/f04035023.png" />-generator group of exponent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040350/f04035024.png" /> (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]]).
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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]]).
  
 
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Revision as of 10:02, 8 September 2014

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) 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). 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). There exist solvable non-Hopfian finitely-generated groups. A finitely-generated residually-finite group (see 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).

References

[1] M.I. Kargapolov, J.I. [Yu.I. Merzlyakov] Merzljakov, "Fundamentals of the theory of groups" , Springer (1979) (Translated from Russian)


Comments

For references see also Finitely-presented group.

How to Cite This Entry:
Finitely-generated group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Finitely-generated_group&oldid=33262
This article was adapted from an original article by Yu.I. Merzlyakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article