21.112 (2026)

A nonempty class $\mathfrak{X}$ of finite groups is said to be complete if $\mathfrak{X}$ is closed under taking subgroups, homomorphic images, and extensions. The symmetric boundary of a complete class $\mathfrak{X}$ other than the class of all finite groups is defined as the largest integer $n$ such that $S_n \in \mathfrak{X}$. Every positive integer $n \neq 3$ coincides with the symmetric boundary of some complete class. It is proved (mod CFSG, D. O. Revin, Algebra i Analiz, 37, no. 1 (2025), 141–176 (Russian)) that, for every complete class $\mathfrak{X}$, there exists a nonnegative integer $m$ with the following property: for every finite group $G$ and each conjugacy class $D$ of $G$, if every $m$ elements of $D$ generate a subgroup belonging to $\mathfrak{X}$, then $\langle D \rangle \in \mathfrak{X}$. The smallest such $m$ is called the Baer–Suzuki width of $\mathfrak{X}$ denoted by $\text{BS}(\mathfrak{X})$. It is also proved (mod CFSG, ibid.) that, for a complete class $\mathfrak{X}$ of symmetric boundary $n$, the value of $\text{BS}(\mathfrak{X})$ is at least $n$ and is bounded above in terms of $n$. For every positive integer $n \neq 3$, let $f_+(n)$ and $f_-(n)$ be respectively the maximum and the minimum of $\text{BS}(\mathfrak{X})$, where $\mathfrak{X}$ runs over all complete classes of symmetric boundary $n$.
$\qquad$ a) Find $f_+(n)$ for $n = 4, 5, 6$. It is known that $f_+(1) = 2, f_+(2) = 3$, and $f_+(n) = 2(n - 1)$ for $n \geqslant 7$.
$\qquad$ b) Is it true that $f_-(n) = n$ for all $n \neq 3$? This is known to be true for $n = 1, 2, 4$.