Focal subgroup theorem

In abstract algebra, the focal subgroup theorem describes the fusion of elements in a Sylow subgroup of a finite group. The focal subgroup theorem was introduced in and is the "first major application of the transfer" according to . The focal subgroup theorem relates the ideas of transfer and fusion such as described in . Various applications of these ideas include local criteria for p-nilpotence and various non-simplicity criteria focussing on showing that a finite group has a normal subgroup of index p. The focal subgroup theorem relates several lines of investigation in finite group theory: normal subgroups of index a power of p, the transfer homomorphism, and fusion of elements. The following three normal subgroups of index a power of p are naturally defined, and arise as the smallest normal subgroups such that the quotient is (a certain kind of) p-group. Formally, they are kernels of the reflection onto the of p-groups (respectively, elementary abelian p-groups, abelian p-groups). Ep(G) is the intersection of all index p normal subgroups; G/Ep(G) is an elementary abelian group, and is the largest elementary abelian p-group onto which G surjects. Ap(G) (notation from ) is the intersection of all normal subgroups K such that G/K is an abelian p-group (i.e., K is an index normal subgroup that contains the derived group ): G/Ap(G) is the largest abelian p-group (not necessarily elementary) onto which G surjects. Op(G) is the intersection of all normal subgroups K of G such that G/K is a (possibly non-abelian) p-group (i.e., K is an index normal subgroup): G/Op(G) is the largest p-group (not necessarily abelian) onto which G surjects. Op(G) is also known as the p-residual subgroup. Firstly, as these are weaker conditions on the groups K, one obtains the containments These are further related as: Ap(G) = Op(G)[G,G]. Op(G) has the following alternative characterization as the subgroup generated by all Sylow q-subgroups of G as q≠p ranges over the prime divisors of the order of G distinct from p.