Vol. 48, No. 2, 1973

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Groups in which Aut(G) is transitive on the isomorphism classes of G

Albert David Polimeni

Vol. 48 (1973), No. 2, 473–480

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.

Mathematical Subject Classification 2000
Primary: 20D45
Received: 7 October 1971
Revised: 13 April 1973
Published: 1 October 1973
Albert David Polimeni