Finitely generated group

A group is finitely generated when it has a finite generating set of a group.

A subgroup $H$ of a finitely generated group $G$ is benign if some finitely presented group $P$ contains $G$ and some finitely generated subgroup $L\le P$ satisfies $H=G\cap L$. The subgroup $H$ itself need not be finitely generated. A centralizing HNN extension of $P$ along $L$ is finitely presented even when the corresponding intersection subgroup $H$ is infinitely generated.

If $N◃G$ is a benign subgroup, use its witness $N=G\cap L$ inside a finitely presented group. Enumerate words on the generators of $L$, together with proofs that each equals a substituted word on the generators of $G$. This enumerates exactly the null words of $G/N$, giving a recursive presentation of a group. The Higman embedding theorem then embeds the finitely generated quotient in a finitely presented group.

For benign subgroups $H,K\le G$, amalgamate their finite-presentation witnesses over the finitely generated $G$, then adjoin independent stable letters $t,s$ centralizing their finite-generation witnesses. Britton's lemma identifies the subgroup generated by $G,t,s$ with the extension centralizing $H,K$. In it, $G\cap G^{ts}=H\cap K$ and $G\cap ⟨G^t,G^s⟩=⟨H,K⟩$. Both conjugate witnesses are finitely generated, proving benignity. For the second identity, the subgroup generated by $G,G^t,G^s$ is the iterated amalgamated free product $G{\ast }_H{G}^t{\ast }_K{G}^s$; its normal forms show that adjoining each outer copy intersects the central copy only in the subgroup already generated by $H,K$.

These upper and lower quantities use $d(H)$, without subtracting one. For a free group $F_n$ with $n\ge 1$, the Nielsen–Schreier formula gives $d(H)/[F_n:H]=n-1+1/[F_n:H]$. The upper value is $n$, attained at the whole group, and the lower value is $n-1$, approached by kernels of maps onto cyclic groups of arbitrarily large finite order. The frequently used shifted rank gradient replaces $d(H)$ by $d(H)-1$ and therefore gives different upper values.

If $G$ has $d$ generators, there are at most $(n!{)}^d$ group homomorphisms $G\to S_n$. Each subgroup of index $n$ is a point stabilizer in a transitive coset group action, so there are at most $n(n!{)}^d$ such subgroups. Consequently finitely many subgroups have index at most any fixed bound. Intersecting them gives a finite-index characteristic subgroup, useful in proving residual finiteness of semidirect products.

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