2 problems

Let $G$ be a finite group. A semicharacter of $G$ is a function $f:G\rightarrow\mathbb{C}^{*}$ satisfying $f(ab)=f(a)f(b)$ whenever $a,b\in G$ commute. The semicharacters form an a…

Odd-prime composition-factor and triple conjecture. The following statements are equivalent: (i) $G$ contains a composition factor whose order is divisible by $pq$; and (ii) $G$ co…