Core (group theory)

In group theory - a branch of mathematics - the term core is used to denote 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.

The normal core

Definition

For a group G, the normal core of a subgroup H is the largest normal subgroup of 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.

$\mathrm {Core} _{S}(H):=\bigcap _{s\in S}{s^{-1}Hs}.$

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.

Significance

Normal cores are important in the context of group actions on sets, where the normal core of the isotropy subgroup on any point acts as the identity on its entire orbit (group theory). 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.

The p-core

Definition

For a prime p, the p-core is defined to be the largest normal p-subgroup in G. It is the normal core of every Sylow p-subgroup of G. The p-core is often denoted $O_{p}(G)$, and in particular appears in the definition 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 $O_{p'}(G)$. 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 $O(G)$. 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.

Significance

The concept of p-cores and p′-cores are important in modular representation theory. 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 subgroup $O_{p',p}(G)$, called 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. This subgroup is defined by $O_{p',p}(G)/O_{p'}(G)=O_{p}(G/O_{p'}(G))$. 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 finite group G is said to be p-constrained for a prime p if $C_{G}(O_{p',p}(G)/O_{p'}(G))\subseteq O_{p',p}(G)$. In particular, every p-soluble and every soluble group is p-constrained.

Solvable Radicals

A related subgroup in concept and notation is the solvable radical. The solvable radical is defined to be the largest solvable normal subgroup, and is denoted $O_{\infty }(G)$. There is some variance in the literature in defining the p′-core of G. A few authors in only a few papers 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

• Aschbacher, M. Finite Group Theory. Cambridge University Press, 2000. ISBN 0-521-78675-4
• Doerk, K. and Hawkes, T. Finite Soluble Groups. Walter de Gruyter, 1992. ISBN 3-110-12892-6
• Huppert, B. and Blackburn, N. Finite Groups II. Springer Verlag, 1982. ISBN 0-387-10632-4