Let G be a finite group and
let A(G) denote the group of automorphisms of G. G is called a T1-group if whenever
L1 and L2 are isomorphic subgroups of G there is a ϕ ∈ A(G) such that L1ϕ= L2. It
is the object of this paper to determine structural properties of T1-groups of odd
order. In particular, if G is a T1-group of odd order, then it is shown that G is a split
extension of a Hall subgroup H, which is a direct product of homocyclic groups, by a
groups K whose Sylow subgroups are cyclic. If G is a supersolvable group of odd
order then it is shown that G is a T1-group if and only if G = HK as in
the previous sentence and K is cyclic with elements which induce power
automorphisms on H. Finally, it is shown that if G is a T1-group of odd
order, then every subnormal subgroup of G is normal if and only if G is
supersolvable.
