在这篇论文，有 | Aut (G) 的有限的组 G：P (G)|在 P (G) 是 G，和 p 的力量自守组的地方， = p 或 pq 被决定， q 是不同素数。特别，我们证明有限的组 G 满足 | Aut (G) ：P (G)|= pq 如果并且仅当 Aut (G)/P (G)}~ S 3 。另外，有限的组的一些另外的班被调查并且分类，它为我们的主要结果的证明是必要的。
A subgroup H of a finite group G is said to be an SS-quasinormal subgroup of G if there is a subgroup B of G such that G = HB and H permutes with every Sylow subgroup of B. In this paper, we investigate the structure of a group under the assumption that every subgroup with order pm of a Sylow p-subgroup P of G is SS-quasinormal in G for a fixed positive integer m. Some interesting results related to the p-nilpotency and supersolvability of a finite group are obtained. For example, we prove that G is p-nilpotent if there is a subgroup D of P with 1 < |D| < |P| such that every subgroup of P with order |D| or 2|D| whenever p = 2 and |D| = 2 is SS-quasinormal in G, where p is the smallest prime dividing the order of G and P is a Sylow p-subgroup of G.
WEI XianBiao1,2 & GUO XiuYun2, 1Department of Mathematics and Physics, Anhui Institute of Architecture and Industry, Hefei 230022, China
