论文标题
将所有元素与普通电源订单相关的所有元素
Groups having all elements off a normal subgroup with prime power order
论文作者
论文摘要
我们考虑一个有限的$ g $，带有普通亚组$ n $，因此$ g \ setminus n $的所有元素都有Prime Power Order。我们证明，如果有一个prime $ p $，以便$ g \ setminus n $中的所有元素都有$ p $ - 功率订单，那么$ g $是$ p $ - group或$ g = pn $，其中$ p $是$ p $ p $ p $ -subgroup和$ $ -subgroup和$（g，p，p，p，p，p，p \ cap n）$是frobenius-wielanius-wielanius-wielanius triple。我们还证明，如果$ g \ setminus n $的所有元素都有主要的电源订单，并且订单可以由两个Primes $ p $和$ q $除外，那么$ g $是$ \ {p，q \} $ - $ g/n $，而$ g/n $是Frobenius组或一个$ 2 $ -FROBENIUS组。如果$ g \ setminus n $的所有要素都有主要的电源订单，并且这些订单至少可以排出三个素数，则$ g $的所有元素都具有Prime Power订单，而$ G/N $则不可征服。
We consider a finite group $G$ with a normal subgroup $N$ so that all elements of $G \setminus N$ have prime power order. We prove that if there is a prime $p$ so that all the elements in $G \setminus N$ have $p$-power order, then either $G$ is a $p$-group or $G = PN$ where $P$ is a Sylow $p$-subgroup and $(G,P,P \cap N)$ is a Frobenius-Wielandt triple. We also prove that if all the elements of $G \setminus N$ have prime power orders and the orders are divisible by two primes $p$ and $q$, then $G$ is a $\{ p, q \}$-group and $G/N$ is either a Frobenius group or a $2$-Frobenius group. If all the elements of $G \setminus N$ have prime power orders and the orders are divisible by at least three primes, then all elements of $G$ have prime power order and $G/N$ is nonsolvable.
