Title: Nilpotent, Controlled and ZJ-controlled Blocks of Finite Groups.

Speaker: An Jianbei (University of Auckland)

Time: Jan 9, 2012 3:00 - 5:00 pm

Venue: Room 82J12 at Jia-Yi-Bing(甲乙丙),BICMR

Abstract: Let G be finite group, and p a prime factor of the order |G| of G. Let B be a p-block of G with a Sylow B-subgroup (D, b) and H a subgroup of G containing D. Then H controls B-fusions if whenever (R, b_R)≤ (D, b), g∈ G and (R, b_R)g≤ (D, b), then g=ch for some c∈ CG(R) and h∈ H.

The block B is nilpotent if D controls B-fusions,B is controlled if NG(D, b) controls B-fusions and B is ZJ-controlled if NG(Z(J(D))) controls B-fusions, where Z(J(D)) is the center of the Thompson subgroup J(D) of D.

Nilpotent blocks were introduced by Broue and Puig in 1980 and controlled blocks were introduced by Alperin and Broue in 1979. In this talk, we are going to discuss recent study on these blocks and in particular, the classifications and applications of these blocks. The classifications of ZJ-controlled blocks implies that the converse of Glauberman's ZJ-theorem holds for all quasi-simple groups.