Topics for Master’s theses
Formanek’s theorem
Let $K$ be a field and $G$ a group. The \emph{zero-divisor conjecture} for group rings asserts that the group ring $K[G]$ is a domain if and only if $G$ is torsion-free. The conjecture has been proven affirmative for several classes of groups. In 1973, Formanek proved it for supersolvable groups.
References:
- T. Y. Lam. A first course in noncommutative rings, volume 131 of Graduate Texts in Mathematics. Springer-Verlag, New York, second edition, 2001.
- D. S. Passman. The algebraic structure of group rings. Robert E. Krieger Publishing Co., Inc., Melbourne, FL, 1985. Reprint of the 1977 original.
Köthe’s conjecture
Köthe’s conjecture is an open problem in ring theory,
formulated in 1930 by Gottfried Köthe.
The conjecture can be formulated in various different ways and
has been shown to be true for various classes of rings,
such as polynomial identity rings and right Noetherian rings.
References:
- J. Krempa. Logical connections between some open problems concerning nil rings. Fund. Math., 76(2):121–130, 1972.
- T. Y. Lam. A first course in noncommutative rings, volume 131 of Graduate Texts in Mathematics. Springer-Verlag, New York, second edition,
The O’Nan-Scott theorem
The result is one of the most influential theorems in the theory of permutation groups. Briefly, the O’Nan-Scott theorem describes maximal subgroups of symmetric groups. Combined with the classification of finite simple groups, there are so many applications that one cannot even count them!
References:
- P. J. Cameron. Finite permutation groups and finite simple groups. Bull. London Math. Soc., 13(1):1–22, 1981.
- J. D. Dixon and B. Mortimer. Permutation groups, volume 163 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1996.
- M. W. Liebeck, C. E. Praeger, and J. Saxl. On the O’Nan-Scott theorem for finite primitive permutation groups. J. Austral. Math. Soc. Ser. A, 44(3):389–396, 1988.
McKay’s conjecture for solvable groups
Counting conjectures play a fundamental role in modern representation theory of finite groups. Among these conjectures, one stands out: McKay’s conjecture. John McKay formulated the problem in 1972. The conjecture is known to be true in several cases, including that of solvable groups.
References:
- G. Navarro. Character theory and the McKay conjecture, volume 175 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2018.
The inverse Galois problem
The Inverse Galois Problem is a fundamental question in mathematics that seeks to determine whether every finite group can be realized as the Galois group of some field extension over the rational numbers. The problem was first explicitly stated by mathematicians in the late 19th and early 20th centuries, notably by David Hilbert. Despite significant progress and partial solutions, the problem remains unsolved for many groups, making it a central topic in algebra and number theory.
References:
- G. Malle and B. H. Matzat. Inverse Galois theory. Springer Monographs in Mathematics. Springer, Berlin, 2018. Second edition.
- J.-P. Serre. Topics in Galois theory, volume 1 of Research Notes in Mathematics. A K Peters, Ltd., Wellesley, MA, second edition, 2008. With notes by Henri Darmon.
Three theorems of Bieberbach
Bieberbach’s theorems are fundamental results in the theory of discrete groups of isometries in Euclidean space. These theorems form the basis for the classification of crystallographic groups, which are crucial in mathematics and physics.
References:
- P. Buser. A geometric proof of Bieberbach’s theorems on crystallographic groups. Enseign. Math. (2), 31(1-2):137–145, 1985.
- L. S. Charlap. Bieberbach groups and flat manifolds. Universitext. Springer-Verlag, New York, 1986.
Frobenius groups and Thompson’s theorem
Suppose a finite group contains a subgroup satisfying specific properties. Using this information, what can be said about the structure of the group itself? A classical and beautiful application of character theory is provided in understanding the structure of the so-called Frobenius groups. In 1960 Thompson proved that Frobenius kernels are nilpotent groups, confirming a long-standing conjecture of Frobenius. Except for some special cases the known proofs for the theorem of Frobenius make use of character theory (or Fourier analysis).
References:
- I. M. Isaacs. Character theory of finite groups. AMS Chelsea Publishing, Providence, RI, 2006. Corrected reprint of the 1976 original Academic Press, New York; MR0460423.
- I. M. Isaacs. Finite group theory, volume 92 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2008.
Wielandt’s automorphism tower theorem
A beautiful theorem of Wielandt states that a the automorphism tower of a finite and centerless stabilizes in finitely many steps. To prove this theorem, subnormality is a pretty crucial idea. This powerful technique is nicely presented in Isaacs book.
References:
- I. M. Isaacs. Finite group theory, volume 92 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2008.
- M. R. Pettet. A note on the automorphism tower theorem for finite groups. Proc. Amer. Math. Soc., 89(1):182–183, 1983.
Groups of central type
A finite group is said to be of central type if it possesses an irreducible complex character which takes the value zero on all non-central elements. (Equivalently, the degree of this character is the square root of the index of the center.)
In the paper of Howlett and Isaacs it is proved that groups of central type must be solvable. This was a conjecture of
Iwahori and Matsumoto.
References:
- R. B. Howlett and I. M. Isaacs. On groups of central type. Math. Z., 179(4):555–569, 1982.
- N. Iwahori and H. Matsumoto. Several remarks on projective representations of finite groups. J. Fac. Sci. Univ. Tokyo Sect. I, 10:129–146 (1964), 1964.
What is a Nichols algebra?
The Nichols algebra of a braided vector space is a braided Hopf algebra. It takes the role of quantum Borel part of a pointed Hopf algebra such as a quantum groups and their well known finite-dimensional analogs.
References:
- I. Heckenberger and H.-J. Schneider. Hopf algebras and root systems, volume 247 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2020.
The Schur cover of a finite group
The Schur multiplier of a group $G$ is the second homology group $H_2(G,\mathbb{Z})$. It was introduced for studying projective representations. Every finite group $G$ has at least one Schur cover $E$. A Schur cover $E$ have the property that every projective representation of $G$ can be lifted to an ordinary representation of $E$.
References:
- D. J. S. Robinson. A course in the theory of groups, volume 80 of Graduate Texts in Mathematics. Springer-Verlag, New York, second edition, 1996.
- J. J. Rotman. An introduction to the theory of groups, volume 148 of Graduate Texts in Mathematics. Springer-Verlag, New York, fourth edition, 1995.