Publications
Extra Special Defect Groups of order p^3 and exponent p
Stuart Hendren, Stuart Hendren Journal of Algebra, 297, 90-97, 2007. In this paper we give results on ($p$-)blocks with the defect groups isomorphic to an extra special group of order $p^{3}.$ We are particularly interested in the number of irreducible ordinary characters, and the number of irreducible Brauer characters in the block. The situation splits naturally into two cases according to the exponent of the extra special group in this paper we concentrate on the exponent $p$ case. We prove that if $p > 7,$ then there exists a non major subsection. We give some conditions which guarantee that Brauer's $k(B)$-conjecture is satisfied. extra special, defect group, group theory, representation theory, block theory, Olsson's Conjecture, Brauer's k(B) conjecture 2007-11-08 09:47:42.0 BibTex
Extra Special Defect Groups of order p^3 and exponent p^2
Stuart Hendren, Look up MR number!! Journal of Algebra, 291, 457-491, 2005. There are many open conjectures regarding $p$-blocks of finite groups. One approach to solving these is to fix the defect group of the $p$-block and attempt to check that the conjectures hold only in this case. This gives further support to the conjecture, more knowledge of that particular case and, hopefully, a better understanding of the general problem. In this paper we give results on ($p$-)blocks with the defect groups isomorphic to an extra special group of order $p^{3}.$ We are particularly interested in the number of irreducible ordinary characters, and the number of irreducible Brauer characters in the block. The situation splits naturally into two cases according to the exponent of the extra special group in this paper we concentrate on the exponent $p^{2}$ case. We prove that Olsson's Conjecture and Brauer's $k(\B)$-Conjecture holds for the exponent $p^{2}$ case. We are also able to calculate two important block invariants ($k(\B) - l(\B)$ and $l(\B) - m_{G, \B}^{(1)}(1)$) in this case. extra special, defect group, group theory, representation theory, block theory, Olsson's Conjecture, Brauer's k(B) conjecture 2008-03-05 09:24:0.0 BibTex
Extra Special Defect Groups
Stuart Hendren, The University of Birmingham, 2004. There are many open conjectures regarding $p$-blocks of finite groups. One approach to solving these is to fix the defect group of the $p$-block and attempt to check that the conjectures hold only in this case. This gives further support to the conjecture, more knowledge of that particular case and, hopefully, a better understanding of the general problem. In this thesis we study ($p$-)blocks with the defect groups isomorphic to an extra special group of order $p^{3}.$ We are particularly interested in the number of irreducible ordinary characters, and the number of irreducible Brauer characters in the block. Our study splits naturally into two cases according to the exponent of the extra special group. We prove that Olsson's Conjecture and Brauer's $k(\B)$-Conjecture hold for the exponent $p^{2}$ case. We are also able to calculate two important block invariants ($k(\B) - l(\B)$ and $l(\B) - m_{G, \B}^{(1)}(1)$) in this case. For the exponent $p$ case we show that the two conjectures hold when certain conditions are fulfilled. In addition we prove that there exists a non-major subsection associated with our block when $p > 7.$ As a consequence we obtain that a finite group with extra special Sylow $p$-subgroups of order $p^{3}$ and $p>7$ has at least two conjugacy classes of $p$-elements. extra special, defect group, group theory, representation theory, block theory, Olsson's Conjecture, Brauer's k(B) conjecture 2008-03-05 09:47:42.0 BibTex