Every topological group is a group retract of a minimal group

We show that every Hausdorff topological group is a group retract of a minimal topological group. This first was conjectured by Pestov in 1983. Our main result leads to a solution of some problems of Arhangel'skii. One of them is the problem about representability of a group as a quotient of a minimal group (Problem 519 in the first edition of ‘Open Problems in Topology’). Our approach is based on generalized Heisenberg groups and on groups arising from group representations on Banach spaces.