Core (group)

In group theory, a branch of mathematics, a core is any of certain special normal subgroups of a group. The two most common types are the normal core of a subgroup and the p-core of a group.

For a group G, the normal core or normal interior of a subgroup H is the largest normal subgroup of G that is contained in H (or equivalently, the intersection of the conjugates of H). 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.

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

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. Thus, in case the action is transitive, the normal core of any isotropy subgroup is precisely the kernel of the action.

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.

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

In this section G will denote a finite group, though some aspects generalize to locally finite groups and to profinite 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. The p-core of G is often denoted $O_{p}(G)$ $O_p(G)$ , 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 $O_{p'}(G)$ $O_{p'}(G)$ . 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 $O(G)$ $O(G)$ . 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 $O_{p',p}(G)$ $O_{p',p}(G)$ is defined by $O_{p',p}(G)/O_{p'}(G)=O_{p}(G/O_{p'}(G))$ $O_{p',p}(G)/O_{p'}(G)=O_{p}(G/O_{p'}(G))$ . For a finite group, the p′,p-core is the unique largest normal p-nilpotent subgroup.

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