Ytukyg

up vote -1 down vote favorite Let $f:Gto G$ be a surjection from a torsion group $G$ onto itself. Let the kernel have infinite cardinality: $lvertker(f)rvert=aleph_0$ What category of function on groups is this? To my mind this cannot be a group automorphism because at the very least an automorphism must map the identity onto itself in order to satisfy $f(acdot b)=f(a)circ f(b)$ . Clearly $f$ does not yield distinct inverses. Is $f$ therefore an endomorphism instead? In part, I'm asking whether an endomorphism on a torsion group can have a kernel with infinite cardinality - what with the elements themselves having finite order and all. I'm asking with half an eye on groups of intermediate growth such as the Grigorchuk group, and any torsion group structure on which variants of the Collatz function might...