Editing Core (group theory)

{{Short description|Any of certain special normal subgroups of a group}}
{{More footnotes needed|date=December 2023}}
In [[group theory]], a branch of [[mathematics]], a '''core''' is any of certain special [[normal subgroup]]s of a [[group (mathematics)|group]]. The two most common types are the '''normal core''' of a [[subgroup]] and the '''''p''-core''' of a group.

==The normal core==
===Definition===
For a group ''G'', the '''normal core''' or '''normal interior'''<ref>Robinson (1996) p.16</ref> of a subgroup ''H'' is the largest [[normal subgroup]] of ''G'' that is contained in ''H'' (or equivalently, the [[intersection (set theory)|intersection]] of the [[conjugate (group theory)|conjugates]] of ''H''). More generally, the core of ''H'' with respect to a [[subset]] ''S''&nbsp;&sube;&thinsp;''G'' is the intersection of the conjugates of ''H'' under ''S'', i.e.
:<math>\mathrm{Core}_S(H) := \bigcap_{s \in S}{s^{-1}Hs}.</math>

Under this more general definition, the normal core is the core with respect to ''S''&nbsp;=&thinsp;''G''. The normal core of any normal subgroup is the subgroup itself.

Dual to the concept of normal core is that of {{em|[[Normal closure (group theory)|normal closure]]}} which is the smallest normal subgroup of ''G'' containing ''H''.

===Significance===
Normal cores are important in the context of [[group action]]s on [[set (mathematics)|sets]], where the normal core of the [[isotropy subgroup]] of any point acts as the identity on its entire [[orbit (group theory)|orbit]]. Thus, in case the action is [[Transitive group action|transitive]], the normal core of any isotropy subgroup is precisely the [[kernel (algebra)|kernel]] of the action.

{{anchor|Core-free}}
A '''core-free subgroup''' is a subgroup whose normal core is the [[trivial subgroup]]. Equivalently, it is a subgroup that occurs as the isotropy subgroup of a transitive, [[Faithful group action|faithful]] group action.

The solution for the [[hidden subgroup problem]] in the [[Abelian group|abelian]] case generalizes to finding the normal core in case of subgroups of arbitrary groups.

==The ''p''-core==
{{redirect|P-core|the computer central processing units|Intel Core#12th generation}}
In this section ''G'' will denote a [[finite group]], though some aspects generalize to [[locally finite group]]s and to [[profinite group]]s.

===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.

===Significance===
Just as normal cores are important for [[Group action (mathematics)|group action]]s on sets, ''p''-cores and ''p''′-cores are important in [[modular representation theory]], which studies the actions of groups on [[vector space]]s. The ''p''-core of a finite group is the intersection of the kernels of the [[simple module|irreducible representation]]s over any field of characteristic ''p''. For a finite group, the ''p''′-core is the intersection of the kernels of the ordinary (complex) irreducible representations that lie in the principal ''p''-block. For a finite group, the ''p''′,''p''-core is the intersection of the kernels of the irreducible representations in the principal ''p''-block over any field of characteristic ''p''.  Also, for a finite group, the ''p''′,''p''-core is the intersection of the centralizers of the abelian chief factors whose order is divisible by ''p'' (all of which are irreducible representations over a field of size ''p'' lying in the principal block).  For a finite, ''p''-constrained group, an irreducible module over a field of characteristic ''p'' lies in the principal block if and only if the ''p''′-core of the group is contained in the kernel of the representation.

===Solvable radicals===
A related subgroup in concept and notation is the solvable radical.  The '''solvable radical''' is defined to be the largest [[solvable group|solvable]] normal subgroup, and is denoted <math>O_\infty(G)</math>.  There is some variance in the literature in defining the ''p''′-core of ''G''.  A few authors in only a few papers (for instance [[John G. Thompson]]'s N-group papers, but not his later work) define the ''p''′-core of an insoluble group ''G'' as the ''p''′-core of its solvable radical in order to better mimic properties of the 2′-core.

==References==
{{reflist}}
{{refbegin}}
* {{ citation | last = Aschbacher | first = Michael | author-link = Michael Aschbacher | title = Finite Group Theory | publisher = [[Cambridge University Press]] | year = 2000 | isbn = 0-521-78675-4 }}
* {{cite book | last1 = Doerk | first1 = Klaus | last2 = Hawkes | first2 = Trevor | title = Finite Soluble Groups | url = https://archive.org/details/finitesolublegro0000doer | url-access = registration | publisher = [[De Gruyter|Walter de Gruyter]] | year = 1992 | isbn = 3-11-012892-6 }}
* {{cite book | last1 = Huppert | first1 = Bertram | last2 =  Blackburn | first2 = Norman | title = Finite Groups II | publisher = [[Springer Science+Business Media|Springer Verlag]] | year = 1982 | isbn = 0-387-10632-4 }}
* {{cite book | title=A Course in the Theory of Groups | volume=80 | series=[[Graduate Texts in Mathematics]] | first=Derek J. S. | last=Robinson | author-link=Derek J. S. Robinson |publisher=[[Springer Science+Business Media|Springer-Verlag]] | year=1996 | isbn=0-387-94461-3 | zbl=0836.20001 | edition=2nd }}
{{refend}}
[[Category:Group theory]]