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Let G be a group. The orbits of the natural action ofAut ( G )on G are called “automorphism orbits” of G , and the number of automorphism orbits of G is denoted byω ( G ). This paper establishes a structure theorem for mixed-order soluble groups of finite rank satisfyingω ( G ) = 4andω ( G ) = 5.

On soluble groups of finite rank with finitely many automorphism orbits