Our aim is to cover foundational material on the theory of characters of finite groups following by the classical textbook “Character Theory of Finite Groups”1 by Isaacs. Some material is referred to G. James and M. Liebeck2 as well.
Schedule
(March 17)Minh-Nguyen Tran, Hoang-Viet Do, and Trung-Nghia Nguyen
Algebras and modules over algebras
Schur’s Lemma and Maschke’s Theorem
Wedderburn’s Theorem
Representations of finite groups
(March 24)Minh-Nguyen Tran
One-to-one correspondence between representations of finite groups and \(FG\)-modules
Class functions and (irreducible) complex characters
The number of irreducible characters
First orthogonality relation
(March 31)Hoang-Viet Do
Inner product of class functions
Second orthogonality relation
Connections between irreducible characters of a finite group and its quotient groups
(April 7)Minh-Nguyen Tran and Hoang-Viet Do
The commutator groups and the number of linear irreducible characters
Connections between the center of a group and the center of characters
Algebraic integers and their basic properties
(April 14)Minh-Nguyen Tran and Hoang-Viet Do
Burnside’s Theorem and the solvability of groups of order \(p^aq^b\)
The integrality of the values of characters
Study integral group bases and Glauberman’s Theorem
(April 21)Minh-Nguyen Tran and Hoang-Viet Do
Tensor products of modules over an arbitrary (non-commutative) ring
Tensor products of \(\mathbb{C}G\)-modules and products of characters
Real characters and Frobenius-Schur Theorem
(April 28)Minh-Nguyen Tran, Hoang-Viet Do, and Trung-Nghia Nguyen
[Linear algebra] Spectral theorem of normal (complex) matrices
Classification of irreducible characters based on the indicator
Characters corresponding to direct products
(May 5)Minh-Nguyen Tran and Hoang-Viet Do
Induced class functions and Frobenius Reciprocity
Imprimitimitivity decomposition of \(FG\)-modules
Monomial characters and M-groups
Taketa’s Theorem on the solvability of \(M\)-groups
(May 12)Minh-Nguyen Tran and Hoang-Viet Do
Permutation characters and doubly transitive actions
Brauer’s Theorem on proper subgroups of a group
Artin’s Theorem on rational-valued characters
(May 19) (online via GMeet) Minh-Nguyen Tran and Hoang-Viet Do
Conjugates of characters and Clifford’s Theorem
Conjugates of representations and Clifford’s Theorem
Correspondence between irreducible characters of a finite groups and irreducible characters of its stabilizer of an irreducible character
Ito’s Theorem on divisibility of index of an abelian group by every degree of irreducible characters
(May 26) (online via GMeet) Minh-Nguyen Tran and Hoang-Viet Do
Gallagher’s Theorem
The restriction of an irreducible character to a normal subgroup of prime index
Necessary and sufficient condition of extendibility of characters
(June 2) (online via GMeet) Minh-Nguyen Tran and Hoang-Viet Do
Frobenius Theorem on Frobenius groups
Brief about linear isometry maps
References
I.M. Isaacs, Character Theory of Finite Groups, AMS-Chelsea, Providence, 2006. ↩
G. James, M. Liebeck, Representations and Characters of Groups (2nd ed.), Cambridge University Press, 2001. ↩