Index of A Subgroup - Normal Subgroups of Prime Power Index

Normal Subgroups of Prime Power Index

Normal subgroups of prime power index are kernels of surjective maps to p-groups and have interesting structure, as described at Focal subgroup theorem: Subgroups and elaborated at focal subgroup theorem.

There are three important normal subgroups of prime power index, each being the smallest normal subgroup in a certain class:

• 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) 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.

As these are weaker conditions on the groups K, one obtains the containments

These groups have important connections to the Sylow subgroups and the transfer homomorphism, as discussed there.