#### Date of Award

12-2020

#### Degree Name

Doctor of Philosophy

#### Department

Mathematics

#### First Advisor

Dr. Clifton Edgar Ealy Jr.

#### Second Advisor

Dr. John Martino

#### Third Advisor

Dr. Jeffry Strom

#### Fourth Advisor

Dr. Justin Lynd

#### Keywords

Abstract algebra, abstract topology, algebraic topology, number theory

#### Abstract

Let *G* be a finite group, and *H* be a subgroup of G. The transfer homomorphism emerges from the natural action of G on the cosets of *H.* The transfer was first introduced by Schur in 1902 [22] as a construction in group theory, which produce a homomorphism from a finite group *G* into *H/H ^{'}* an abelian group where

*H*is a subgroup of

*G*and

*H'*is the derived group of

*H*. One important first application is Burnside’s normal

*p*-complement theorem [5] in 1911, although he did not use the transfer homomorphism explicitly to prove it.

**Burnside Theorem.** *Let G be a finite group*, *and let P be a sylow p-subgroup that is contained in the center of its normalizer, then G has a normal subgroup H which has elements of P as its coset representatives.*

Emil Artin in 1929 [1] Extended the definition to the situation *G* is infinite and *H* is a subgroup of *G* of finite index. The first place the transfer homomorphism appeared in a textbook was in 1937, Lehrbuch der Gruppentheorie, by Hans Zassenhaus [28]. In 1959 the transfer was popularized in the mathematical community in America by Marshall Hall in the textbook The Theory of Groups [14] where the transfer "in German *Die Verlagerung"* as a homomorphism from a group *G* into a subgroup *H* of finite index was introduced.

In an effort to define the transfer homomorphism for profinite groups, Oliver Schirokauer in 1996 [21] published a paper in which he presented a new definition for the standard cohomological transfer as an integral.

In this study we will give a definition of the transfer homomorphism for profinite groups which is an analog of Marshall Hall’s definition of the transfer homomorphism for finite groups. Our definition depends on the group action and structure, in relation to axiom of choice and ordinal numbers, using the permanent map.

#### Access Setting

Dissertation-Open Access

#### Recommended Citation

Shatnawi, Mohammad, "On the Local Theory of Profinite Groups" (2020). *Dissertations*. 3691.

https://scholarworks.wmich.edu/dissertations/3691