Let p be a prime and let G be a finite group such that the smallest prime that divides |G| is p. We find sharp bounds, depending on p, for the commuting probability and the average character degree to guarantee that G is nilpotent or supersolvable.

中文翻译：

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通勤概率和平均特征度方面的超解性和幂零性

设p为素数，G为有限群，使得整除 | 的最小素数 G | 是p。我们根据p找到通勤概率和平均特征度的尖锐界限，以保证G是幂零的或超可解的。