有限群的最小子群覆盖

Minimum Covering by Subgroups of Finite Group

设p为素数,G是非循环有限群,群G的最小子群覆盖所包含的子群个数记为n(G),群G的最小循环子群覆盖所包含的子群个数记为nc(G),群G的最小Abel子群覆盖所包含的子群个数记为na(G),则3≤n(G)≤|G|-1,nc(Cp×…×Cp)m个=pm-1+…+p+1(m≥2),nc(Cpr×Cp)=r(p-1)+2(r≥1),na(Cpr×Cps)=p+1(r≥s≥1).

Abstract： Let p be a prime, and G be a non-cyclic finite group. We denote by n （G） the number of subgroups of minimum coverings by subgroups of G , and denote by nc （G） the number of subgroups of minimum coverings by cyclic subgroups of G , and denote by na （G） the number of subgroups of minimum coverings by Abelian subgroups of G , then （1）3≤ n（G）≤｜G｜-1, （2）nc（Cp×…×Cp）}m个=p^m-1＋…＋p＋1,where m≥2, （3）nc（Cp^r × Cp） = r（p- 1） ＋2,where r ≥ 1, （4）na（Cp^r × Cp^s） = p ＋ 1,where r ≥ s ≥1.