Some of the applications of character theory are the proofs of Burnside $p^aq^b$ theorem, Frobenius theorem and factorization of the group determinant (the problem which led Frobenius to character theory).
I would like to know what are some of the other problems which has been solved by the application of character theory especially from number theory and group theory.
Representations of finite groups can be used to prove Hurwitz's 1,2,4,8 theorem. See http://www.math.uconn.edu/~kconrad/blurbs/linmultialg/hurwitzrepnthy.pdf.
@Alexander Gruber - A celebrated conjecture of Dedekind asserts that for any finite algebraic extension $K$ of $\mathbb{Q}$, the zeta function $\zeta_K(s)$ is divisible by the Riemann zeta function $\zeta(s)$. That is, the quotient $\zeta_K (s)/\zeta(s)$ is entire. More generally, Dedekind conjectures that if $L$ is a finite extension of $K$, then $\zeta_L(s)/\zeta_K(s)$ should be entire. This conjecture is still open, I believe.
By the work of Aramata and Brauer the conjecture is known to be true if $L/K$ is Galois. If $L$ is contained in a solvable extension of $K$, then Uchida and van der Waall have independently proved Dedekind’s conjecture. Proofs rely on properties of M-groups and hence this is a very nice application of character theory to algebraic number theory!
Character theory provides a better language to talk about certain group theoretic problems. Here is an example.
Definition. Let $G$ be a finite group.
- If $M\unlhd H \leqslant G$ and $H/M$ is cyclic, we call $(H,M)$ a pair.
- For $g\in G$ and $H\leqslant G$, write $$F_H(g)=\{[g,h]:h\in H\cap H^{g^{-1}}\}$$ If $(H,M)$ is a pair and $F_H(g)\not\subseteq M$ for all $g\in G$, we say that $(H,M)$ is a good pair.
- If $(H,M)$ and $(K,L)$ are good pairs, we say that they are related in $G$ if there exists a $g\in G$ for which $H^g\cap L=K \cap M^g$. It can be proven that being related in $G$ is an equivalence relation on the set of good pairs in $G$, and we denote by $m_G$ the number of equivalence classes under this relation.
- Denote by $n_G$ the number of equivalence classes of elements of $G$ under the relation $x\sim y \Leftrightarrow \langle x \rangle \text{ is conjugate to } \langle y \rangle\text{ in }G$.
- Let $\mathcal{M}$ be the class of finite groups with the property that $m_G=n_G$.
Goodness, that's a long definition. $\mathcal{M}$ is clearly a very difficult class of groups to study, impossible to understand. All the pieces fit together in some strange way, but it isn't quite clear what it all means.
Consider the following alternative definition.
Definition. A character of a finite group $G$ is monomial if it is induced from a linear character of a subgroup of $G$. We say $G$ is in the class $\mathcal{M}$ if every irreducible character of $G$ is monomial.
The groups in both these definitions are called $\mathcal{M}$-groups, with proof of equivalence given yonder. We know a lot of things about them: $\mathcal{M}$-groups are solvable, supersolvable groups are $\mathcal{M}$-groups, normal subgroups of $\mathcal{M}$-groups and their quotients are both $\mathcal{M}$-groups, every solvable group can be embedded in an $\mathcal{M}$-group.
It is not at all obvious that these definitions are equivalent. The first definition was introduced $55$ years after the second, motivated by informal questions posed by several people who studied $\mathcal{M}$-groups using pairs. The purpose of this post is not to devalue the study of pairs in $\mathcal{M}$-groups, but to emphasize that character theory provides an easier language in which to study some problems in group theory, even when it is not logically necessary as in the case of Burnside's and Frobenius' theorems.
One of the prominent applications of representation theory of the symmetric group is to the study of random walks on groups. Look for instance Chapter 11 in 'Representation Theory of Finite Groups' by Benjamin Steinberg.
The Feit–Thompson theorem, or Odd Order Theorem, states that every finite group of odd order is solvable. It was proved by Walter Feit and John Thompson (1963). Its proof ($2^8-1$ pages long) relies heavily on character theory.
