Class groups for integral representations of metacyclic groups

Periodic Projective Resolutions

“…For Type I, the key result available is due to Galovitch, Reiner, and Ullom [13]. Let p be an odd prime, q a divisor of (p -1), and r a primitive qth root of 1 (mod p).…”

Class groups of metacyclic groups of order p'q,p a regular prime

“…It is clear that k r has a unique maximal ideal p r and that k r /p r £ k; thus we have a decomposition U(k r ) £ U(k)® U(k r , p r ). Proposition 3.2 shows that U(k r , p r ) lies in the image of U(R r ), and an argument similar to that given in [3,Lemma 3.2] shows that coker (U(R r ) -> U(k)) is cyclic of order q/(q, 2). Remark 3.4.…”

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Let p be an odd prime, let q be a divisor of p -1, and let a be a primitive <j-th root modulo p. For each natural number r the metacyclic group G r is defined by G r = <h n f\ hf = 1 = fjhj-1 = V>.Our aim is to give an inductive description of the class group C(ZG r ) which generalizes that given in [3] for the case r = 1. Let L r be the cyclotomic field of p r -th roots of unity, let K r be unique subfield of L r such that (L r : K r ) = q, and let R r be the algebraic integers of K r .

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“…Our aim is to give an inductive description of the class group C(ZG r ) which generalizes that given in [3] for the case r = 1. Let L r be the cyclotomic field of p r -th roots of unity, let K r be unique subfield of L r such that (L r : K r ) = q, and let R r be the algebraic integers of K r .…”

A CHARACTERIZATION OF THE UNIT GROUP IN ℤ[T×C2]

“…Furthermore, Higman gave a general structure theorem for U (ZA), where A is a finite abelian group. Other results include: A 4 and S 4 by Allen-Hobby [1,2], D 2p by Passman-Smith [21], G = C p ⋊ C q , where q is a prime dividing p − 1 by Galovitch-Reiner-Ullom [4], |G| = p 3 by RitterSehgal [23], and U (ZS 3 ) by Hughes-Pearson [7]. Jespers and Parmenter [10] gave a more explicit description of U (ZS 3 ).…”