Composition factors of groups and factors of their order

Krings, Marvin; Niemeyer, Alice Catherine (Thesis advisor); Glasby, Stephen (Thesis advisor)

Aachen : RWTH Aachen University (2022)
Dissertation / PhD Thesis

Dissertation, RWTH Aachen University, 2022


Let \(T\) be a finite simple group. For a finite group \(G\), we determine an upper bound on the number \(c_T(G)\) of composition factors of \(G\) that are isomorphic to \(T\). We consider this problem for non-abelian \(T\) as well as for abelian \(T\), and we obtain similar results for both cases. Our bound is in \(\mathcal{O}(n)\) for a permutation group \(G \le \Sym(n)\) and in \(\mathcal{O}(\log n)\) if, in addition, \(G\) is primitive, quasi primitive or semiprimitive. The approach to proving these results involves induction on the permutation degree, using the O'Nan-Scott Theorem if \(G\) is primitive If \(G \le \GL_d(q)\), then our bound is in \(\mathcal{O}(d)\). Similar to the permutation group case, we prove it by describing \(G\) in terms of matrix groups of smaller dimension and permutation groups. We use Aschbacher's Theorem to find such a description. Then we apply the bound inductively and apply the bound for the permutation group case to prove the bound for \(G \le \GL_d(q)\). As a consequence of our bound, we get that if \(c_T(G) > 0\), then \(T\) can be embedded into \(\PGL_d(q')\) for some prime power \(q'\). In most cases, we even have that \(q'\) and \(q\) are powers of the same prime. Using this observation, we describe all the simple groups that may occur as composition factors of \(G \le \GL_d(q)\) if \(d \le 12\). Furthermore, we investigate which groups \(G \le \GL_d(q)\) have particularly large values of \(c_T(G)\) in relation to \(d\). More precisely, we show that \(c_T(G) < \frac{d}{2}\) in most cases and describe the exceptions to this bound. The second main topic of this thesis is bounding the \(p\)-part \(\lvert G \rvert_p\) of the order of a primitive permutation group \(G \le \Sym(n)\) that is almost simple. That is, we assume \(\Inn(T) \triangle left eq G \le \Aut(T)\) for a non-abelian finite simple group \(T\). The order of \(\Aut(T)\) is known by the Classification of the Finite Simple Groups and given for example in the Atlas of finite groups. For each \(G\) in that Classification, we deduce a bound for \(\nu_p(G) = \log_p \lvert G \rvert_p\) in terms of \(n\). We get that \(\nu_p(G) = \mathcal{O}(\sqrt{n})\) and in most cases \(\nu_p(G) = \mathcal{O}(\log n)\).