Theory of Finite Groups is a powerful tool in algebraic combinatorics. It uses group representations to study group structure through character tables, which capture essential information about a group's irreducible representations and conjugacy classes.
This theory connects to Combinatorial by providing methods to analyze and count solutions to equations in groups. It showcases how abstract algebra concepts can solve concrete combinatorial problems, bridging pure and applied mathematics.
Character tables of finite groups
Definition and properties of character tables
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A character of a representation is the trace of the corresponding
The character is a class function on the group, meaning it is constant on conjugacy classes
The character table of a group is a table whose rows correspond to the irreducible characters and whose columns correspond to the conjugacy classes of the group
The entry in the ith row and jth column is the value of the ith irreducible character on an element of the jth conjugacy class
The character table of a determines the group up to isomorphism
Two groups with the same character table are isomorphic (e.g., D8 and Q8)
Computing character tables and their applications
The sum of the squares of the dimensions of the irreducible representations equals the order of the group
This follows from the for irreducible characters
The number of irreducible representations of a group equals the number of conjugacy classes of the group
This is because the character table is a square matrix
Character tables can be used to study the structure of a group, such as determining its center, commutator subgroup, and normal subgroups
For example, the center of a group consists of elements g for which ∣χ(g)∣=χ(1) for all irreducible characters χ
Orthogonality relations for characters
First and second orthogonality relations
The first orthogonality relation states that the sum of the products of the values of two different irreducible characters over all group elements is zero
In other words, distinct irreducible characters are orthogonal with respect to the inner product defined by summing over the group
The second orthogonality relation states that for an irreducible character χ, the sum of ∣χ(g)∣2 over all group elements g equals the order of the group
This implies that an irreducible character has norm equal to the square root of the order of the group
Applications of orthogonality relations
Orthogonality relations can be used to decompose a given character into a sum of irreducible characters
The multiplicity of an irreducible character in this decomposition can be computed using the inner product of characters
The orthogonality relations imply that the irreducible characters form an orthonormal basis for the space of class functions on the group
This is with respect to the inner product defined by summing over the group
Orthogonality relations are crucial in proving character-theoretic formulas for counting solutions to equations in groups
For example, they are used in the proof of Burnside's lemma (also known as the Cauchy-Frobenius lemma)
Character-theoretic formulas for counting solutions
Burnside's lemma and its applications
Burnside's lemma is a formula for counting the number of orbits of a group action
It states that the number of orbits equals the average number of fixed points over all group elements
The number of fixed points of a group element g acting on a set X can be computed using the character of the permutation representation associated to the action
Specifically, the number of fixed points equals the value of this character at g
Applying Burnside's lemma to the conjugation action of a group on itself yields a formula for the number of conjugacy classes
The number of conjugacy classes equals the average value of an irreducible character over the group
Counting solutions to equations in groups
Character-theoretic methods can be used to count the number of solutions to certain equations in finite groups
For example, the number of solutions to the equation xn=1 in a group G equals the sum of the values of the irreducible characters of G at an element of order n
Similar techniques can be applied to count the number of solutions to other equations, such as x2=1 or xy=yx
These methods often involve expressing the number of solutions in terms of character sums and then using orthogonality relations to simplify the expressions
Conjugacy classes and normal subgroups using characters
Determining conjugacy classes from character tables
Two elements of a group are conjugate if and only if they have the same character values for all irreducible characters
Thus, the conjugacy classes can be determined from the character table
The kernel of a character χ is the set of group elements g for which χ(g)=χ(1)
This is a normal subgroup of the group
If χ is a faithful character (i.e., its kernel is trivial), then the center of the group consists precisely of the elements g for which ∣χ(g)∣=χ(1)
Characterizing normal subgroups using characters
A subgroup H of a group G is normal if and only if every irreducible character of G restricts to a sum of irreducible characters of H
This criterion can be used to test for normality using the character tables of G and H
The commutator subgroup of a group (i.e., the subgroup generated by all commutators) is the intersection of the kernels of all linear characters (i.e., one-dimensional characters) of the group
This characterization can be used to compute the commutator subgroup from the character table
Characters can also be used to prove that certain subgroups are characteristic (i.e., invariant under all automorphisms of the group)
For example, the center and the commutator subgroup are always characteristic subgroups