--- abstract: 'Given a finite group $G$, let ${\mathop{\mathrm{Cl}}\nolimits}(G)$ denote the set of isomorphism classes of irreducible $kG$-modules. For $s$ a prime number, the $s$th [*Fitting invariant*]{} $F_s(G)$ is defined to be the number of classes of ${\mathop{\mathrm{Cl}}\nolimits}(G)$ whose order is divisible by $s$. In this paper, we examine the existence of universal finite groups $G$ with respect to given cardinal numbers of their Fitting invariants, and determine the maximal number of classes of ${\mathop{\mathrm{Cl}}\nolimits}(G)$ whose order is divisible by a given prime $s$, for every $s \geq 3$. We also determine the minimal number of classes of ${\mathop{\mathrm{Cl}}\nolimits}(G)$ whose order is divisible by a given prime $s$, and the minimal size of a set $\Sigma$ such that every prime $s$ can be written as the sum of $|G|$ and elements of $\Sigma$.' author: - 'Michael A. Dokuchaev[^1]' - 'Alexei E. Zalesskii[^2]' title: | The Fitting invariant of a finite group\ and groups of small size --- Introduction {#introduction .unnumbered} ============ Let $G$ be a finite group. For any prime number $s$, we denote by $o(G,s)$ the number of isomorphism classes of irreducible $kG$-modules with Fitting invariant $s$, i.e., the number of isomorphism classes of irreducible $kG$-modules $U$ satisfying ${\mathop{\mathrm{End}}\nolimits}_G(U) \cong k[X]/(X^s)$ (see Subsection \[subsect:Fitting\_inv\]). The Fitting invariant of a finite group has appeared in various contexts in representation theory and invariant theory. For example, it plays an important role in the theory of rationality of finite groups (see [@Golubov-Karpilovsky Chapter 6]). A result of [@Guralnick-Robinson] states that the number of isomorphism classes of irreducible $kG$-modules of a finite group $G$ is at most $\frac{1}{2} |G| (|G|-1) \prod_{p \, \mid \, |G|} \frac{p}{p-1}$, where the product is over all primes $p$ dividing $|G|$. By this theorem, there exist only finitely many groups with $o(G,1) > 0$, i.e., $o(G,1)$ is a well-defined integer. However, the corresponding results for $o(G,s)$, with $s \geq 2$, do not hold for arbitrary finite groups (see [@Liebeck-Shalev] for further details). The Fitting invariant of a finite group $G$ has been studied from the point of view of group theory. By a result of Schur [@Schur], if $G$ is a non-abelian simple group, then $o(G,s)$ is positive for every prime number $s$. If $G$ is not solvable, then $o(G,2)$ is either $1$ or $0$ [@Dolfi-Ragusa-Tamburini], so every finite solvable group has the property that $o(G,s)$ is positive for exactly one $s$. A criterion for a finite group to have this property is given by [@Dolfi-Ragusa], and the set $\{{\mathop{\mathrm{Cl}}\nolimits}(G) \mid G \mbox{ is a finite group}\}$ has been studied in [@Guralnick-Thompson]. More generally, in the present paper we examine the existence of universal finite groups $G$ with respect to given cardinalities of $o(G,s)$, and determine their maximal and minimal values. Given a prime number $s \geq 3$, let $c_s(G)$ be the maximal number of classes of ${\mathop{\mathrm{Cl}}\nolimits}(G)$ whose order is divisible by $s$. We prove that, for every $s \geq 3$, there exists a universal finite group $G$ for which $o(G,s) = c_s(G)$ (see Theorem \[thm:c\_s\_for\_all\_s\]). Thus, in order to determine $c_s(G)$, it suffices to consider universal finite groups $G$ with respect to a given number of distinct primes $s$. We prove that $c_s(G) = 1$ for all $s$ if and only if every irreducible $kG$-module with $s$-length is absolutely irreducible. This occurs, in particular, when $G$ is a non-abelian simple group or a Frobenius group (see Lemma \[lem:o(G,s)=1\], and Lemma \[lem:o(G,s)=1\_when\_simple\_or\_Frobenius\]). For a Frobenius group $G$, we show that $c_3(G) = c_5(G) = c_6(G) = 1$ (see Theorem \[thm:5-6-7\]), but the general picture remains unclear. We also study the case $s=3$ (see Theorems \[thm:d-adic\] and \[thm:3-adic\]), where the following result is proven: for any non-cyclic finite group $G$, we have $c_3(G) \geq 3$. We give a characterization of all finite groups $G$ satisfying $c_3(G) = 3$ (see Theorem \[thm:3-adic\_for\_groups\_with\_size\_3\]), and we provide some infinite classes of finite groups satisfying $c_3(G) \geq 3$ (see Examples \[exa:c\_3=3\_cyclic\], \[exa:c\_3=3\_non-cyclic\], and \[exa:c\_3=3\_non-cyclic\_for\_2-groups\]); in particular, we prove that if $G$ is a Frobenius group whose Frobenius kernel is of prime power order and whose Frobenius complement is cyclic, then $c_3(G) = 3$ (see Example \[exa:c\_3=3\_for\_Frobenius\]). It is interesting to compare the results for $c_s(G)$ with the work of Dolfi and Ragusa [@Dolfi-Ragusa] who studied the number of isomorphism classes of irreducible modules of a finite group $G$ whose minimal dimension is divisible by a prime $s$. We prove that $c_s(G) \leq o(G,s)$ (see Proposition \[prop:o(G,s)\]); in particular, if $c_s(G) > 0$, then $c_s(G) \leq o(G,s)$. It is known that every finite group has a subgroup $H$ such that $o(H,s) = o(G,s)$ (see [@Guralnick-Thompson]), so in particular every finite group has a subgroup $H$ such that $c_s(H) = c_s(G)$. We prove that, for any prime $s$, there exist groups $G$ such that $c_s(G) = |G|$ and $o(G,s) = \frac{1}{2} |G| (|G|-1) \prod_{p \, \mid \, |G|} \frac{p}{p-1}$ (see Theorems \[thm:s=3\] and \[thm:s=5\]), and there exist groups $G$ such that $c_s(G) \geq |G|$ and $o(G,s) = 1$ (see Theorem \[thm:s=3\]). A criterion for a group to have $c_s(G) = |G|$ is given by Theorem \[thm:s=3\_general\], which shows that every finite non-solvable group has $c_3(G) = |G|$ if and only if $G$ is either a Frobenius group of prime degree or a non-abelian simple group (see Theorem \[thm:s=3\_general\]). On the other hand, we prove that $c_s(G) \geq 2^{|G|}$ for any prime $s \geq 7$ (see Theorem \[thm:s=2\_7\_and\_8\]), and we construct finite groups $G$ with $c_3(G) = 2$ and $o(G,s) \geq |G|$ for any $s \geq 3$ (see Example \[exa:2-groups\]), thereby completing the proof of a result of [@Dolfi-Ragusa], which states that, for $s \geq 3$, there exists a universal finite group $G$ with $c_s(G) = 2$. It is known that there exist universal finite groups $G$ such that $o(G,s) = 2^{|G|}$ for some $s \