# Finitely-generated group

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).