Local Conjugations of Groups and Applications to Number Fields
This dissertation studies pairs of subgroups H, H' of a finite group G together with a bijective map Φ H −> H' that is a local conjugation, meaning that each element h in H is conjugate in G to its image Φ(h). The map Φ is not required to take products to products. The motivation for studying such pairs comes from a paper of F. Gassmann in 1926, in which he formulated an equivalent but different-sounding condition now known as Gassmann’s condition. There are now at least ten equivalent reformulations of Gassmann’s condition, of which local conjugation is perhaps the most elementary. The utility of studying local conjugation is that it raises natural questions. For example, if we do additionally require that the map Φ preserve products (that is, if it is required that Φ be an isomorphism as well as a local conjugation), does it follow that Φ is a global conjugation? An example showing the answer is no is given in this dissertation. Many applications of local conjugacy have been discovered. In number theory, the groups H, H', G appear as Galois groups of field extensions of the field of algebraic number field k, and H, H' are locally conjugate in G if and only if the fixed fields K, K' of H, H' have identical Dedekind zeta functions. In 1985, Sunada looked at H, H', G as groups of deck isometrices of coverings of Riemann surfaces and showed that when H, H' are locally conjugate but not conjugate in G then the corresponding Riemann surfaces are isospectral but non-isometric. And more recently, locally conjugate subgroups of a finite group G have been used to produce pairs of nonisomorphic graphs with identical Ihara zeta functions. All of this motivated the study of local conjugacy in this dissertation. Among other things, yet another reformulation, called cycle number equivalence, was discovered, which gives as a corollary a new proof of a theorem of Stuart and Perlis.
Bloc Equivalence, Arithmetical Equivalence, Gassmann's Condition
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