Is the following statement true: Any surjective homomorphism $f:G\to H$ of groups with equal rank maps every minimal generating system $x$ of $G$ to a minimal generating system $y$ of $H$? "Minimal generating set" means "generating set of minimal cardinality". Is it only valid for finite rank? How about infinite rank?



Surjective group homomorphism sending minimal generating system to minimal generating system
I'm assuming that "minimal generating set" means "a generating set that does not properly contain a generating set" (that is, every proper subset does not generate). In that reading, the statement is false in both finite and infinite ranks.


For a counterexample in finite rank, let $G$ be infinite cyclic generated by $x$ (rank $1$), and $H$ be cyclic of order $2$ generated by $y$ (also rank $1$). Let $f\colon G\to H$ be the map sending $x$ to $y$. The minimal generating set $\{x^2,x^3\}$ (which generates $G$, but no proper subset does) is mapped to $\{e,y\}$, which is not minimal. Or you could take $H$ to be cyclic of order $4$, if you want to avoid the trivial element, as then you get $y^2$ and $y^3$; which is not minimal.


For infinite rank, take $G=H$ be a direct sum of countably infinitely many copies of the infinite cyclic group; take $\{x\_i\}\_{i=1}^{\infty}$ as a basis for $G$, and $\{y\_j\}\_{j=1}^{\infty}$ as a basis for $H$. Take $f\colon G\to H$ be the map that sends $x\_{2i-1}$ to $y\_i$ and $x\_{2i}$ to $y\_i^{-1}$. 


---


If by "minimal generating set" you mean "generating set of minimal cardinality", then given that the rank of a group is the least cardinality of a generating set, then the answer is positive: under a surjective homomorphism, a generating set is mapped to a generating set. So if $X$ is a generating set for $G$ with $\mathrm{rank}(G)$ elements, then $f(X)$ is a generating set for $H$ with at most $\mathrm{rank}(G)=\mathrm{rank}(H)$ elements, and since any generating set has at least $\mathrm{rank}(H)$ elements, it follows that $f(X)$ has exactly $\mathrm{rank}(H)$ elements (in cardinality) and so is a generating set of minimal cardinality.


