On the source algebra equivalence class of blocks with cyclic defect groups, I

4 Topological Equivalence of Linear Representations for Cyclic Groups : I

In the two parts of this paper we prove that the Reidemeister torsion invariants determine topological equivalence of G–representations, for G a finite cyclic group.

On blocks with abelian defect groups of small rank

Let B be a p-block of a finite group with abelian defect group D. Suppose that D has no elementary abelian direct summand of order p. Then we show that B satisfies Brauer’s k(B)-Conjecture (i. e. k(B) ≤ |D|). Together with former results, it follows that Brauer’s k(B)-Conjecture holds for all blocks of defect at most 3. We also obtain some related results.

On defect groups for generalized blocks of the symmetric group

In a paper of 2003, B. Külshammer, J. B. Olsson and G. R. Robinson defined l-blocks for the symmetric groups, where l > 1 is an arbitrary integer. In this paper, we give a definition for the defect group of the principal l-block. We then check that, in the Abelian case, we have an analogue of one of M. Broué’s conjectures.

On the Automorphism Groups and Equivalence of Cyclic Combinatorial Objects

We determine the permutation groups that arise as the automorphism groups of cyclic combinatorial objects. As special cases we classify the automorphism groups of cyclic codes. We also give the permutations by which two cyclic combinatorial objects on p elements are equivalent.

18.703 Modern Algebra, Cyclic Groups