{"product_id":"automorphism-orbits-and-element-orders-in-finite-groups-almost-solubility-and-the-monster","title":"Automorphism Orbits and Element Orders in Finite Groups: Almost-Solubility and the Monster","description":"For a finite group G, we denote by ?(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) := ?(G) ? o(G) and q(G) := ?(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 ?(G) = o(G)). We show that the index |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 characterisation of the Fischer-Griess Monster group M.","brand":"American Mathematical Society","offers":[{"title":"Default Title","offer_id":54260740981080,"sku":"9781470465445","price":85.99,"currency_code":"EUR","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0278\/1295\/4195\/files\/9781470465445_ba3c6520-afbc-49de-a9f4-715bf15ff646.jpg?v=1777888842","url":"https:\/\/agendabookshop.com\/products\/automorphism-orbits-and-element-orders-in-finite-groups-almost-solubility-and-the-monster","provider":"Agenda Bookshop","version":"1.0","type":"link"}