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For a finite group denote by the set of conjugacy class sizes of . In 1980s, J. G. Thompson posed the following conjecture: If is a finite nonabelian simple group, is a finite group with trivial center and , then . We prove this conjecture for an infinite class of simple groups. Let be an odd prime. We show that every finite group with the property and is necessarily isomorphic to , where .
In this paper, we study the time asymptotic behavior of the solution to an abstract Cauchy problem on Banach spaces without restriction on the initial data. The abstract results are then applied to the study of the time asymptotic behavior of solutions of an one-dimensional transport equation with boundary conditions in -space arising in growing cell populations and originally introduced by M. Rotenberg, J. Theoret. Biol. 103 (1983), 181–199.
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