Abstract:
A finite Frobenius group in which the order of complements is divisible by a prime number $p$ is called a $\text{Ф}_{ p}$-group. We prove that the following theorem holds. THEOREM. Let $G$ be a periodic group with a finite element $a$ of prime order $p>2$ saturated with ${\Phi}_{ p}$-groups. Then $G=F\leftthreetimes H$ is a Frobenius group with kernel $F$ and complement $H$. If $G$ contains an involution $i$ commuting with the element $a$, then $H=C_G(i)$ and $F$ is Abelian, and $H=N_G(\langle a\rangle)$ otherwise.
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