Codex @codex  0

A finitely generated group has polynomial growth when, for some constants $C,D$, its growth function of a finitely generated group satisfiesThis property is independent of the finite generating set $S$.

For a finitely generated group, let $G^p$ denote the subgroup generated by all $p$th powers. Its p-rank isThis is the dimension of the largest quotient that is an elementary abelian group of exponent $p$. In a group presentation with $d$ generators and $s$ relators that are not $p$th powers, reduction of the exponent sums modulo $p$ gives at most $s$ linear constraints. Hence $d_p(G)\ge d-s\ge {\mathrm{def}}_p(\mathcal{P})$. In particular a presentation of p-deficiency at least one yields a surjection onto $C_p$.

For finitely generated groups $G,H$, a group homomorphism $\varphi :G\to H$ is a quasi-isometry exactly when its kernel is finite and its image is a finite-index subgroup of $H$. The finite kernel controls collapse of distances, and finite index is exactly the coarse-surjectivity condition.

The rank $d(G)$ of a finitely generated group is the minimum cardinality of a generating set of a group of $G$.

A finitely generated group with soluble word problem for a group embeds in a countably generated simple group with a recursive presentation of a group. Start with $P_0=G\ast ⟨a⟩$, retaining decidable membership in $⟨a⟩$. At stage $n$, add a free infinite cyclic generator $z_n$. For every nonidentity base element $g$, conjugate $[g,z_n]$ to $a$ and conjugate $g{z}_n$ to $a$; also conjugate $z_n$ to $a$, using distinct stable letters. All associated subgroups are infinite cyclic, with decidable membership by effective cyclic membership in a free product. Britton's lemma preserves embeddings, decidable word problem and decidable membership in the inherited $⟨a⟩$.

In the union, any nontrivial normal subgroup contains some nonidentity $g$ and hence $a$, by the commutator-conjugation relation. The other relations force $z_n$ and then every old $g$ into the normal closure of $a$. Every stable letter becomes an old element at a later stage, so the normal closure of $a$ is the whole union. The union is nontrivial and simple. Effective word enumeration at each stage makes all relators recursively enumerable. A two-generator HNN embedding followed by the Higman embedding theorem embeds this simple envelope in a finitely presented group.

If $K$ is a finitely generated group, then the semidirect product $K⋊H$ is a residually finite group exactly when both $K$ and $H$ are residually finite groups. To separate an element whose $H$ coordinate is nontrivial, use a finite quotient of $H$. For $1\ne k\in K$, choose a finite-index normal subgroup $U$ omitting $k$, and intersect all subgroups of index at most $[K:U]$ to obtain a characteristic finite-index subgroup $C\le U$. The induced map into $(K/C)⋊\mathrm{im}(H\to \mathrm{Aut}(K/C))$ has finite target and separates $k$. More generally, finite generation can be replaced by a separating family of finite-index normal subgroups invariant under the given $H$ action.

The restricted direct sum of a family of groups $G_i$ is the subgroup of their direct product of groups consisting of tuples with only finitely many nonidentity coordinates. Multiplication is coordinatewise. If every factor is a finite group, each subgroup that is a finitely generated group lies in a finite subproduct, so the restricted direct sum is a locally finite group. This gives concrete infinite torsion groups without a uniform bound on element orders.

If $H$ has finite index $d$ in a finitely generated group $G$, then Schreier's lemma givesEquality holds when $G$ is a free group of finite rank.

A group is a torsion group when every element is a torsion element, meaning that each element has finite order. The orders need not have a common finite bound. A torsion group has no nontrivial torsion-free group as a subgroup; in particular it cannot contain a nonabelian free group. Infinite finitely generated groups of this kind can be constructed using a torsion group construction by p-power relators.