Bounding the order of complex linear groups and permutation groups with selected composition factors
In this interesting article, the author gets some general bounds on the order of finite subgroups of \(\mathrm{GL}(n, \mathbb{C})\). The work is originally motivated by some question of P. Etingof (related to growth rates of tensor powers in symmetric tensor categories, see [\textit{K. Coulembier} et al., Pure Appl. Math. Q. 20, No. 3, 1141--1179 (2024; Zbl 1537.18020)]). The generality of the methods used lead the reviewer to provide a description of the main concepts developed and the main results obtained by the author.\N\NLet \(G\) be a finite group. A section of \(G\) is a group of the form \(H/K\) where \(H \leq G\) and \(K \trianglelefteq H\), a section \(G/H\) is normal if \(H \trianglelefteq G\). Let \(\mathbf{P}\) be a property of finite groups which is inherited by all sections (hereditary for brevity) and we let \(\mathbf{Q}\) be a property which is inherited by all normal sections (normally hereditary for brevity). A normally hereditary property \(\mathbf{Q}\) is extant if at least one finite group has property \(\mathbf{Q}\) and an extant property \(\mathbf{Q}\) is irredundant if there is at least one finite group which does not have property \(\mathbf{Q}\). If \(\mathbf{Q}\) is an extant normally hereditary property of finite groups and \(G\) is finite let\N\(A(\mathbf{Q})= \{ |G : F(G)|^{1/(n-1)} \mid n \geq 2, G \leq \mathrm{GL}(n,\mathbb{C}) \text{ and } G \text{ has } \mathbf{Q} \} \),\N\(B(\mathbf{Q})= \{ |G|^{1/(n-1)} \mid n \geq 2, G \leq \mathrm{Sym}(n) \text{ and } G \text{ has } \mathbf{Q} \} \),\N\(C(\mathbf{Q})= \{ |G : A|^{1/(n-1)} \mid n \geq 2, G \leq \mathrm{GL}(n,\mathbb{C}) \text{ and } G \text{ has } \mathbf{Q} \} \),\Nwhere \(A\) is a maximal normal subgroup of the finite group \(G\).\N\NThe first general result, which makes use of the \textsf{CFSG}, is Theorem 1.1: Let \(\mathbf{P}\) be an extant irredundant hereditary property of finite groups. Then the sets \(A(\mathbf{P})\), \(B(\mathbf{P})\) and \(C(\mathbf{P})\) are all bounded above. Hence, letting \(\alpha(\mathbf{P})\), \(\beta(\mathbf{P})\) and \(\gamma(\mathbf{P})\) denote the least upper bounds of \(A(\mathbf{P})\), \(B(\mathbf{P})\) and \(C(\mathbf{P})\), respectively, we have: (i) Whenever \(n\) is a positive integer and \(G\) is a finite subgroup with property \(\mathbf{P}\) of \(\mathrm{GL}(n,\mathbb{C})\), then we have \([G:F(G)]\leq \alpha(\mathbf{P})^{n-1}\). Furthermore, \(\alpha(\mathbf{P})\) is the smallest real number with this property. (ii) Whenever \(n\) is a positive integer and \(G\) is a subgroup with property \(P\) of the symmetric group \(\mathrm{Sym}(n)\), then we have \(|G| \leq \beta(\mathrm{Sym}(n))^{n-1}\). Furthermore, \(\beta(P)\) is the smallest real number with this property. (iii) Whenever \(n\) is a positive integer and \(G\) is a finite subgroup with property \(\mathbf{P}\) of \(\mathrm{GL}(n,\mathbb{C})\), then \(G\) has an abelian normal subgroup \(A\) with \(|G : A| \leq \gamma(P)^{n-1}\). Furthermore, \(\gamma(\mathbf{P})\) is the smallest real number with this property.\N\NTheorem 1.1 is a consequence of Theorem 1.2: Let \(\mathbf{Q}\) be an extant normally hereditary property of finite groups. Then the sets \(A(\mathbf{Q})\), \(B(\mathbf{Q})\) and \(C(\mathbf{Q})\) are all bounded above if and only if there are only finitely many alternating groups with property \(\mathbf{Q}\). Furthermore, when these sets are all bounded above and we let \(m\) denote the largest integer such that \(\mathrm{Alt}(m)\) has property \(\mathbf{Q}\), then we have: (i) if \(m \leq 151\), then \N\[\max \{\alpha(\mathbf{Q}), \beta(\mathbf{Q}), \gamma(\mathbf{Q}) \} \leq 60,\] \N(ii) if \(m > 151\), then\N\[\N\left( \frac{m!}{2} \right)^{1/(m-1)} \leq \min \{\alpha(\mathbf{Q}), \beta(\mathbf{Q}), \gamma(\mathbf{Q}) \}\N\]\Nand\N\[\N\max \{\alpha(\mathbf{Q}), \beta(\mathbf{Q}), \gamma(\mathbf{Q}) \} \leq ( m! )^{1/(m-2)}.\N\]\NThe questions of P. Etingof give two situations when Theorem 1.1 may be applied to obtain explicit bounds. Let \(p\) be a prime and let the property \(\mathbf{P}\) be the property of having order prime to \(p\), then \(\mathbf{P}\) is clearly hereditary, extant and irredundant. For \(p\) an odd prime, \(p-1\) is the largest integer \(m\) such that \(\mathrm{Alt}(m)\) has property \(\mathbf{P}\), while for \(p=2\), only the alternating groups \(\mathrm{Alt}(2)\) and \(\mathrm{Alt}(3)\) have property \(\mathbf{P}\). If instead \(\mathbf{P}\) is the property of having abelian Sylow \(p\)-subgroups, then the largest integer \(m\) such that the alternating group \(\mathrm{Alt}(m)\) has property \(\mathbf{P}\) is \(p^{2}-1\) if \(p\) is odd and \(m=5\) if \(p=2\). Hence, Theorem 1.2 is applied, obtaining (see Corollary 1.5): Let \(p\) be a prime and \(G\) be a finite subgroup of \(\mathrm{GL}(n, \mathbb{C})\) of order prime to \(p\), then \(G\) has an abelian normal subgroup \(A\) such that if \(p \leq 151\), then \N\[|G:A| \leq 60^{n-1}\] \Nand if \(p>151 \), then \N\[|G:A| \leq \big( (p-1)!\big)^{\frac{n-1}{p-3}}.\N\]\NAnd (see Corollary 1.7): Let \(H\) be a finite subgroup of \(\mathrm{GL}(n,\mathbb{C})\). If \(H\) has any non-solvable alternating composition factor, let \(m \geq 5\) be the largest integer such that \(\mathrm{Alt}(m)\) is a composition factor of \(H\). Then either \(H\) has an abelian normal subgroup of index at most \(60^{n-1}\), or else \(n+1 \geq m > 151\) and \(H\) has an abelian normal subgroup \(A\) with\N\[\N\frac{m!}{2} \leq |H : A| \leq \big (m! \big)^{\frac{n-1}{m-2}} \leq \big (n+1 \big)!\N\]\NThe paper under review contains many other results of the same kind which are too long to report here.
scientific article; zbMATH DE number 7876018