Evans, John Derek Peter; (2018) Group algebras of metacyclic type. Doctoral thesis (Ph.D), UCL (University College London).

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Abstract

Let Λ = Z[G] denote the integral group ring of a finite group G. In the first part of this thesis we consider the stable syzygies Ω_r(Z) over Λ. These are defined to be the stable classes of the intermediate modules in a free Λ-resolution of the trivial module Z. If we let p be an odd prime, then the groups of concern to us will be G_1 = D_2p which has free period 4, and G_2 = C_p ⋊ C_3 which has free period 6. Along the way it will also be necessary to consider the syzygies of the cyclic group C_n which has free period 2, the smallest possible nontrivial periodic resolution. The key point of note in each of these cases is that the augmentation ideal splits, thereby allowing us to show the existence of a diagonalised resolution. Moreover, there exist two strands corresponding to the action of the generators of C_p, and of either C_2 or C_3. For each strand we show there exists a group structure within the stable class generated by part of the first syzygy Ω_1(Z), and in which part of the zeroth syzygy Ω_0(Z) is the identity. In the second part of this thesis we make the jump to infinite groups. By setting G = C_p⋊C_q where p, q are prime such that q|p−1, we discuss the stably free modules over Z[G×F_n], where F_n denotes the free group of rank n. As we shall see, the stably free modules over this group ring are necessarily trivial; that is, they are free.

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