Prüfer rank

In mathematics, especially in the area of algebra known as group theory, the Prüfer rank of a pro-p group measures the size of a group in terms of the ranks of its elementary abelian sections. The rank is well behaved and helps to define analytic pro-p-groups. The term is named after Heinz Prüfer. In mathematics, especially in the area of algebra known as group theory, the Prüfer rank of a pro-p group measures the size of a group in terms of the ranks of its elementary abelian sections. The rank is well behaved and helps to define analytic pro-p-groups. The term is named after Heinz Prüfer. The Prüfer rank of pro-p-group G {displaystyle G} is where d ( H ) {displaystyle d(H)} is the rank of the abelian group where Φ ( H ) {displaystyle Phi (H)} is the Frattini subgroup of H {displaystyle H} . As the Frattini subgroup of H {displaystyle H} can be thought of as the group of non-generating elements of H {displaystyle H} , it can be seen that d ( H ) {displaystyle d(H)} will be equal to the size of any minimal generating set of H {displaystyle H} . Those profinite groups with finite Prüfer rank are more amenable to analysis. Specifically in the case of finitely generated pro-p groups, having finite Prüfer rank is equivalent to having an open normal subgroup that is powerful. In turn these are precisely the class of pro-p groups that are p-adic analytic - that is groups that can be imbued with a p-adic manifold structure.