Editing P-group (section)

===Non-trivial center===
One of the first standard results using the [[class equation]] is that the [[Center (group theory)|center]] of a non-trivial finite ''p''-group cannot be the trivial subgroup.<ref>[[Conjugacy class#Example|proof]]</ref>

This forms the basis for many inductive methods in ''p''-groups.

For instance, the [[normalizer]] ''N'' of a [[proper subgroup]] ''H'' of a finite ''p''-group ''G'' properly contains ''H'', because for any [[counterexample]] with ''H'' = ''N'', the center ''Z'' is contained in ''N'', and so also in ''H'', but then there is a smaller example ''H''/''Z'' whose normalizer in ''G''/''Z'' is ''N''/''Z'' = ''H''/''Z'', creating an infinite descent. As a corollary, every finite ''p''-group is [[nilpotent group|nilpotent]].

In another direction, every [[normal subgroup]] ''N'' of a finite ''p''-group intersects the center non-trivially as may be proved by considering the elements of ''N'' which are fixed when ''G'' acts on ''N'' by conjugation. Since every central subgroup is normal, it follows that every minimal normal subgroup of a finite ''p''-group is central and has order ''p''. Indeed, the [[socle of a group|socle]] of a finite ''p''-group is the subgroup of the center consisting of the central elements of order ''p''.

If ''G'' is a ''p''-group, then so is ''G''/''Z'', and so it too has a non-trivial center. The preimage in ''G'' of the center of ''G''/''Z'' is called the [[Center (group theory)#Higher centers|second center]] and these groups begin the [[upper central series]]. Generalizing the earlier comments about the socle, a finite ''p''-group with order ''p<sup>n</sup>'' contains normal subgroups of order ''p<sup>i</sup>'' with 0 ≤ ''i'' ≤ ''n'', and any normal subgroup of order ''p<sup>i</sup>'' is contained in the ''i''th center ''Z''<sub>''i''</sub>. If a normal subgroup is not contained in ''Z''<sub>''i''</sub>, then its intersection with ''Z''<sub>''i''+1</sub> has size at least ''p''<sup>''i''+1</sup>.