Abstract
The p-rank of a finite p-group is the maximal rank of an abelian subgroup. For odd primes the p-groups of p-rank at most 2 are classified by Blackburn. We use this classification in order to prove Olsson’s Conjecture for all blocks with defect groups of p-rank at most 2 provided p > 3. We also develop general methods which deal with controlled blocks.
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References
An, J.: Dade’s conjecture for the Tits group. New Zealand J. Math. 25(2), 107–131 (1996)
An, J.: The Alperin and Dade conjectures for Ree groups \(^{2}F_{4}(q^{2})\) in non-defining characteristics. J. Algebra 203(1), 30–49 (1998)
An, J.: Controlled blocks of the finite quasisimple groups for odd primes. Adv. Math. 227(3), 1165–1194 (2011)
An, J., Eaton, C.W.: Blocks with extraspecial defect groups of finite quasisimple groups. J. Algebra 328, 301–321 (2011)
An, J., O’Brien, E.A.: The Alperin and Dade conjectures for the O’Nan and Rudvalis simple groups. Commun. Algebra 30(3), 1305–1348 (2002)
An, J., O’Brien, E.A., Wilson, R.A.: The Alperin weight conjecture and Dade’s conjecture for the simple group J 4. LMS J. Comput. Math. 6, 119–140 (2003)
Blackburn, N.: On a special class of p-groups. Acta Math. 100, 45–92 (1958)
Conway, J.H., Curtis, R.T., Norton, S.P., Parker, R.A., Wilson, R.A.: ATLAS of finite groups. Oxford University Press, Eynsham (1985). Maximal subgroups and ordinary characters for simple groups, With computational assistance from J.G. Thackray
Díaz, A., Ruiz, A., Viruel, A.: All p-local finite groups of rank two for odd prime p. Trans. Am. Math. Soc. 359(4), 1725–1764 (2007)
Guralnick, R.M.: Commutators and commutator subgroups. Adv. Math. 45(3), 319–330 (1982)
Hendren, S.: Extra special defect groups of order p 3 and exponent p. J. Algebra 313(2), 724–760 (2007)
Héthelyi, L., Külshammer, B., Sambale, B.: A note on olsson’s conjecture. J. Algebra 398, 364–385 (2014)
Huppert, B.: Endliche Gruppen. I. Die Grundlehren der Mathematischen Wissenschaften, Band 134. Springer, Berlin (1967)
Isaacs, I.M.: Finite Group Theory. Graduate Studies in Mathematics, vol. 92. American Mathematical Society, Providence (2008)
Kessar, R., Stancu, R.: A reduction theorem for fusion systems of blocks. J. Algebra 319(2), 806–823 (2008)
Narasaki, R., Uno, K.: Isometries and extra special Sylow groups of order p 3. J. Algebra 322(6), 2027–2068 (2009)
Ruiz, A., Viruel, A.: The classification of p-local finite groups over the extraspecial group of order p 3 and exponent p. Math. Z. 248(1), 45–65 (2004)
Sambale, B.: Further evidence for conjectures in block theory. Algebra Number Theory 7(9), 2241–2273 (2013)
Stancu, R.: Control of fusion in fusion systems. J. Algebra Appl. 5(6), 817–837 (2006)
Uno, K.: Conjectures on character degrees for the simple Thompson group. Osaka J. Math. 41(1), 11–36 (2004)
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Sambale, B. (2014). Defect Groups of p-Rank 2. In: Blocks of Finite Groups and Their Invariants. Lecture Notes in Mathematics, vol 2127. Springer, Cham. https://doi.org/10.1007/978-3-319-12006-5_11
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DOI: https://doi.org/10.1007/978-3-319-12006-5_11
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