Motivated by recent results on the minimal base of a permutation group, we introduce a new local invariant attached to arbitrary finite groups. More precisely, a subset \(\Delta \) of a finite group G is called a p-base (where p is a prime) if \(\langle \Delta \rangle \) is a p-group and \(\mathrm {C}_G(\Delta )\) is p-nilpotent. Building on results of Halasi–Maróti, we prove that p-solvable groups possess p-bases of size 3 for every prime p. For other prominent groups, we exhibit p-bases of size 2. In fact, we conjecture the existence of p-bases of size 2 for every finite group. Finally, the notion of p-bases is generalized to blocks and fusion systems.

根据排列组的最小基数的最新结果，我们引入了附加到任意有限组的新局部不变性。更确切地说，子集\（\德尔塔\）有限群G ^称为p -基（其中，p是质数），如果\（\ langle \德尔塔\ rangle \）是一个p -基团和\（\ mathrm {C} _G（\ Delta）\）为p-幂。在Halasi-Maróti的成果的基础上，我们证明了p -solvable组具有p大小3 -bases每一个素数p。对于其他重要群体，我们展示p-大小为2的p-基。实际上，我们推测每个有限组的大小为2的p-基的存在。最后，p-基的概念被推广到块和融合系统。