Subgroup theorem for valuated groups and the CSA property

A valuated group with normal forms is a group with an integer-valued length function satisfying some of Lyndon's axioms (Lyndon, 1963 [Lyn63], ) and an additional axiom considered by Hurley (1980) [Hur80]. We prove a subgroup theorem for valuated groups with normal forms analogous to Grushko–Neumann's theorem. We also study the CSA property in such groups.