满足Gaschütz补定理的群
Groups satisfying Gaschütz's complement theorem
AI总结：
该研究刻画了使Gaschütz补定理在去掉阿贝尔假设后仍成立的有限群N，补充了第二作者的前期工作。
AI中文摘要：
Gaschütz的经典定理指出，当有限群G的阿贝尔正规子群N在满足N≤H≤G的某个子群H中有补，且gcd(|N|, [G:H])=1时，N在G中有补。我们刻画了使该定理在去掉阿贝尔假设后仍成立的有限群N：当且仅当Z(N)∩N'=1，且N在全形Hol(N)被N'的对角副本的商群中有补时成立。这完成了第二作者之前的一篇论文。
英文摘要：
A classical theorem of Gaschütz states that an abelian normal subgroup N of a finite group G has a complement in G whenever it has a complement in some subgroup H with $N \leq H \leq G$ and gcd(|N|, [G:H]) = 1. We characterize the finite groups N for which this theorem remains valid without the abelian hypothesis: this is the case if and only if $Z(N) \cap N' = 1$ and N has a complement in a quotient of the holomorph Hol(N) by a diagonal copy of N'. This completes a previous paper by the second author.
