The two most common types are the normal core of a subgroup and the p-core of a group.
More generally, the core of H with respect to a subset S ⊆ G is the intersection of the conjugates of H under S, i.e.
Normal cores are important in the context of group actions on sets, where the normal core of the isotropy subgroup of any point acts as the identity on its entire orbit.
The solution for the hidden subgroup problem in the abelian case generalizes to finding the normal core in case of subgroups of arbitrary groups.
For a prime p, the p-core of a finite group is defined to be its largest normal p-subgroup.
It is the normal core of every Sylow p-subgroup of the group.
, 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
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
For a finite group, the p′,p-core is the unique largest normal p-nilpotent subgroup.
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.
Just as normal cores are important for group actions on sets, p-cores and p′-cores are important in modular representation theory, which studies the actions of groups on vector spaces.
The p-core of a finite group is the intersection of the kernels of the irreducible representations 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.
A related subgroup in concept and notation is the solvable radical.
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.