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"For a finite group G, we denote by [omega](G) the number of Aut(G)-orbits on G, and by o(G) the number of distinct element orders in G. In this paper, we are primarily concerned with the two quantities d(G) :[equals] [omega](G) - o(G) and q(G) :[equals] [omega](G)/ o(G), each of which may be viewed as a measure for how far G is from being an AT-group in the sense of Zhang (that is, a group with [omega](G) [equals] o(G)). We show that the index [absolute value]G : Rad(G) of the soluble radical Rad(G) of G can be bounded from above both by a function in d(G) and by a function in q(G) and o(Rad(G)). We also obtain a curious quantitative characterization of the Fischer-Griess Monster group M"--
Finite groups. --- Automorphisms. --- Group theory and generalizations -- Abstract finite groups -- Arithmetic and combinatorial problems. --- Group theory and generalizations -- Abstract finite groups -- Finite simple groups and their classification. --- Group theory and generalizations -- Abstract finite groups -- Automorphisms.
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"Let G be a group. An automorphism of G is called intense if it sends each subgroup of G to a conjugate; the collection of such automorphisms is denoted by Int(G). In the special case in which p is a prime number and G is a finite p-group, one can show that Int(G) is the semidirect product of a normal p-Sylow and a cyclic subgroup of order dividing p 1. In this paper we classify the finite p-groups whose groups of intense automorphisms are not themselves p-groups. It emerges from our investigation that the structure of such groups is almost completely determined by their nilpotency class: for p 3, they share a quotient, growing with their class, with a uniquely determined infinite 2-generated pro-p group"--
Finite groups. --- Automorphisms. --- Nilpotent groups. --- Group theory and generalizations -- Abstract finite groups -- Nilpotent groups, $p$-groups. --- Group theory and generalizations -- Abstract finite groups -- Automorphisms. --- Group theory and generalizations -- Special aspects of infinite or finite groups -- Automorphism groups of groups. --- Group theory and generalizations -- Structure and classification of infinite or finite groups -- Limits, profinite groups. --- Group theory and generalizations -- Structure and classification of infinite or finite groups -- Automorphisms of infinite groups.
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