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Core (group theory)
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===Significance=== Just as normal cores are important for [[Group action (mathematics)|group action]]s on sets, ''p''-cores and ''p''β²-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''β²-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''β²,''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''β²,''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''β²-core of the group is contained in the kernel of the representation.
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