关于极小非幂零群的正规Sylow子群的换位子群的生成元集

On a Set of Generate Elements of the Commuatator Subgroup of the Normal Sylow Subgroup of a Minimal Non-nilpotent Group

讨论了极小非幂零群的正规Sylow子群P的换位子群P′,确定了换位子群P′的一个生成元集.从而用简单的和纯群论方法得到换位子群P′的阶｜P′｜的上界,并进而得到不等式｜P′｜≤｜P｜1/3.此外,通过相关的本原单位根,给出了换位子群P′的阶｜P′｜达到这个上界的一个必要条件.

The author discusses the commutator subgroup P′ of the normal Sylow subgroup P of a minimal non-nilpotent group and determine a set of generate elements of the commutator subgroup P′. Using this simple and group-theorical way, the author gets an above limit of the order|P′| of the commutator subgroup P′, and then gets the inequality |P′|≤3|P|. In addition, in terms of a related primitive root of unity, he gives a necessary condition under which the order |P′| of the commutator subgroup P′ is this above limit.