Editing Core (group theory) (section)

===Definition===
For a [[prime number|prime]] ''p'', the '''''p''-core''' of a finite group is defined to be its largest normal [[p-group|''p''-subgroup]].  It is the normal core of every [[Sylow subgroup|Sylow p-subgroup]] of the group.  The ''p''-core of ''G'' is often denoted <math>O_p(G)</math>, and in particular appears in one of the definitions of the [[Fitting subgroup]] of a [[finite group]].  Similarly, the '''''p''′-core''' is the largest normal subgroup of ''G'' whose order is coprime to ''p'' and is denoted  <math>O_{p'}(G)</math>.  In the area of finite insoluble groups, including the [[classification of finite simple groups]], the 2′-core is often called simply the '''core''' and denoted <math>O(G)</math>.  This causes only a small amount of confusion, because one can usually distinguish between the core of a group and the core of a subgroup within a group.  The '''''p''′,''p''-core''', denoted <math>O_{p',p}(G)</math> is defined by <math>O_{p',p}(G)/O_{p'}(G) = O_p(G/O_{p'}(G))</math>. For a finite group, the ''p''′,''p''-core is the unique largest normal ''p''-nilpotent subgroup.

The ''p''-core can also be defined as the unique largest subnormal ''p''-subgroup; the ''p''′-core as the unique largest subnormal ''p''′-subgroup; and the ''p''′,''p''-core as the unique largest subnormal ''p''-nilpotent subgroup.

The ''p''′ and ''p''′,''p''-core begin the '''upper ''p''-series'''.  For sets ''π''<sub>1</sub>, ''π''<sub>2</sub>, ..., ''π''<sub>''n''+1</sub> of primes, one defines subgroups O<sub>''π''<sub>1</sub>, ''π''<sub>2</sub>, ..., ''π''<sub>''n''+1</sub></sub>(''G'') by:
:<math>O_{\pi_1,\pi_2,\dots,\pi_{n+1}}(G)/O_{\pi_1,\pi_2,\dots,\pi_{n}}(G) = O_{\pi_{n+1}}( G/O_{\pi_1,\pi_2,\dots,\pi_{n}}(G) )</math>
The upper ''p''-series is formed by taking ''π''<sub>2''i''−1</sub> = ''p''′ and ''π''<sub>2''i''</sub> = ''p;'' there is also a [[lower p-series|lower ''p''-series]].  A finite group is said to be '''''p''-nilpotent''' if and only if it is equal to its own ''p''′,''p''-core.  A finite group is said to be '''''p''-soluble''' if and only if it is equal to some term of its upper ''p''-series; its '''''p''-length''' is the length of its upper ''p''-series.  A finite group ''G'' is said to be '''[[p-constrained]]''' for a prime ''p'' if <math>C_G(O_{p',p}(G)/O_{p'}(G)) \subseteq O_{p',p}(G)</math>.

Every nilpotent group is ''p''-nilpotent, and every ''p''-nilpotent group is ''p''-soluble.  Every soluble group is ''p''-soluble, and every ''p''-soluble group is ''p''-constrained.  A group is ''p''-nilpotent if and only if it has a '''normal ''p''-complement''', which is just its ''p''′-core.