Ghost Numbers of Group Algebras

The quaternion group has ghost number three

“…By [4], the ghost number is at least three. So we have to show that every threefold ghost M f − → N is stably trivial.…”

“…In [4], Christensen and Wang studied ghost numbers for p-group algebras. They gave conjectural upper and lower bounds for the ghost number of an arbitrary p-group, and also showed that the ghost number (over a field of characteristic two) of the quaternion group Q 8 is either three or four.…”

Ghost Numbers of Group Algebras II

“…And since P is a p-group, the radical length of a projective-free kP -module L is equal to the length of L with respect to the projective class (F u , G u ) [15,Proposition 4.5]. It follows that, for M ∈ stmod(kG) and L ∈ stmod(kP ), if M↓ P and L are projective-free modules, then len S u (M) = rad len(M↓ P ) and rad len(L) = len S u (L↑ G ).…”

“…Still assuming that the principal block is generated by k, we show that the ghost number of kG is greater than or equal to the ghost number of kP , by first showing that the composite of inducing up from P to G followed by projection onto the principal block is faithful. In Section 4.3, working at the prime 2, we show that for a dihedral group D 2ql of order 2ql, with q a power of 2 and l odd, the principal block is generated by k and the ghost number of kD 2ql is equal to the ghost number of the Sylow 2-subgroup D 2q , which was shown to be q 2 + 1 in [15]. By computing the simple ghost lengths of modules in non-principal blocks, we are also able to show that the simple ghost number of kD 2ql is again q 2 + 1 .…”

“…Some preliminary work was done in [11], where the ghost numbers of cyclic p-groups were computed, and various upper and lower bounds were obtained in other cases. Substantial progress was made in our previous paper [15], where we computed the ghost numbers of k(C 3 × C 3 ) and other algebras of wild representation type, as well as the ghost numbers of dihedral 2-groups, the first non-abelian computations.…”