21.134 (2026)

For a finite group $G$, let the type of $G$ be the function on positive integers whose value at $n$ is the number of solutions of the equation $x^n = 1$ in $G$.
$\qquad$ a) Is it true that a group having the same type as a group with trivial solvable radical must also have trivial solvable radical? Note that there are solvable and nonsolvable groups with the same type (see 12.37).
$\qquad$ b) Is it true that a group having the same type as an almost simple group must be isomorphic to it? This is true for a group having the same type as a simple group, as follows from the affirmative answer to 12.39.