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In this paper, we determine the TI subgroups of the simple Suzuki groups Sz(q). More generally, we determine those nontrivial subgroups that are disjoint from some of their conjugates. It turns out that the latter are exactly those subgroups that have ordinary depth 3. The Sylow 2-subgroups of simple Suzuki groups belong to the class of so-called Suzuki 2-groups, which have been studied extensively by Higman. These results were extended later by Goldschmidt, Shaw, Shult, Gross, Wilkens and Bryukhanova. As a corollary of our investigations, we get some interesting results for the Sylow 2-subgroups of Suzuki groups, as well. We relate this to an open problem on Suzuki 2-groups, and we ask a question concerning that. We also give some characterization of Suzuki groups. [ABSTRACT FROM AUTHOR]

We determine the combinatorial and the ordinary depth of the maximal subgroups of the simple Ree groups 2G2(q). As an application of these calculations, we determine the subdegrees of primitive actions of the groups 2G2(q). We also improve an earlier estimate of Burness, Liebeck, and Shalev on the base size of" non-standard" primitive actions of 2G2(q) on the coset space of maximal subgroups from 3 to 2. [ABSTRACT FROM AUTHOR]

Let G be a finite group, let p be a prime divisor of the order of G and let k(G) be the number of conjugacy classes of G. By disregarding at most finitely many non-solvable p-solvable groups G, we have $k(G)\geq2\smash{\sqrt{p-1}}$ with equality if and only if if $\smash{\sqrt{p-1}}$ is an integer, $G=C_{p}\rtimes\smash{C_{\sqrt{p-1}}}$ and CG(Cp) = Cp. This extends earlier work of Héthelyi, Külshammer, Malle and Keller. [ABSTRACT FROM PUBLISHER]

We prove that a finite solvable group whose order is divisible by a prime p has at least conjugacy classes. 2000 Mathematics Subject Classification 20C15, 20D10, 20F16. [ABSTRACT FROM AUTHOR]