# Composition factors of groups and factors of their order

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

Dissertation, RWTH Aachen University, 2022

Abstract

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 $\nu_p(G) = \mathcal{O}(\log n)$.

Institutions

• Department of Mathematics 
• Algebra Teaching and Research Area