8.2 Character Theory of Finite Groups
5 min read•july 30, 2024
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., and )
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 for which for all irreducible characters
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 , the sum of over all group elements 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 acting on a set 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
- 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 in a group equals the sum of the values of the irreducible characters of at an element of order
- Similar techniques can be applied to count the number of solutions to other equations, such as or
- 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 is the set of group elements for which
- This is a normal subgroup of the group
- If is a faithful character (i.e., its kernel is trivial), then the center of the group consists precisely of the elements for which
Characterizing normal subgroups using characters
- A subgroup of a group is normal if and only if every irreducible character of restricts to a sum of irreducible characters of
- This criterion can be used to test for normality using the character tables of and
- 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
Key Terms to Review (18)
Abelian Group: An abelian group is a set equipped with an operation that combines two elements to produce a third element, satisfying four key properties: closure, associativity, the existence of an identity element, and the existence of inverses. Additionally, the operation is commutative, meaning the order of the elements does not affect the outcome. This structure is fundamental in various mathematical areas, including character theory, where understanding group properties helps analyze representations and their symmetries.
Burnside's Theorem: Burnside's Theorem is a fundamental result in group theory that provides a way to count the number of distinct objects under a group of symmetries by calculating the average number of points fixed by the group actions. This theorem is crucial in combinatorics as it connects symmetry with counting, allowing one to find the number of orbits in a set when acted upon by a group.
Character: In mathematics, particularly in representation theory, a character is a homomorphism from a group to the multiplicative group of complex numbers. This means that characters provide a way to study the structure of a group through its representations, allowing for an understanding of how group elements act on vector spaces. Characters are crucial for analyzing representations because they encapsulate important information about the group's structure and its symmetries.
Dim(v): In the context of character theory of finite groups, dim(v) refers to the dimension of a representation space associated with a character v. This dimension indicates the number of basis vectors needed to describe the representation, highlighting the structure and complexity of the associated group actions. The concept is critical in understanding how group representations can vary and how they relate to the group's characters.
Endomorphism: An endomorphism is a type of function that maps a mathematical structure to itself, preserving the operations defined on that structure. In the context of group theory, an endomorphism is a homomorphism from a group to itself, allowing for an analysis of the group's internal structure and symmetries. This concept is vital for understanding the character theory of finite groups, as it provides insights into how representations can be manipulated and classified.
Finite group: A finite group is a set equipped with a binary operation that satisfies the group axioms (closure, associativity, identity, and invertibility), where the number of elements in the set is finite. This concept is essential in understanding many structures in algebra, particularly in analyzing symmetries and group actions, as well as in character theory which studies the representation of these groups through linear transformations.
Group Actions: Group actions are mathematical ways in which a group can be represented as symmetries or transformations that act on a set. This concept is vital because it helps to analyze how groups operate on different structures, revealing their underlying properties and relationships. By understanding group actions, we can explore important topics like orbits, stabilizers, and the relationship between group theory and geometry.
Group Algebra: A group algebra is a mathematical structure that combines elements of a group with coefficients from a field, allowing for the construction of linear combinations of group elements. This concept connects algebraic structures with representation theory, enabling the study of group actions in a more manageable way through linear algebra. Group algebras are essential in understanding characters and representations of finite groups, and they play a significant role in the development of Hopf algebras as well.
Irreducible Representation: An irreducible representation of a group is a homomorphism from the group to the general linear group of a vector space that has no proper subrepresentation. This means that the only invariant subspaces under the action of the group are trivial (the zero space and the whole space), making it a building block for understanding how groups can act on vector spaces. Irreducible representations play a crucial role in character theory, allow for the classification of representations, and are essential in analyzing complex representations through simpler ones.
Linear character: A linear character is a homomorphism from a group to the multiplicative group of a field, often the complex numbers, that reflects the structure of the group in a one-dimensional way. This type of character can be thought of as a representation of the group that preserves the group operation and has a particularly simple form, making it easier to study properties such as irreducibility and induction in character theory.
Matrix representation: Matrix representation refers to the use of matrices to represent linear transformations and other algebraic structures, often simplifying complex operations in mathematical contexts. In the study of character theory for finite groups, matrix representations help in understanding how group elements can be represented as linear transformations on vector spaces, revealing deep connections between algebra and geometry.
Module: A module is a mathematical structure that generalizes vector spaces by allowing scalars to come from a ring instead of a field. This structure captures the essence of linear algebra while extending it to contexts where the usual properties of fields may not apply, making modules essential in areas such as representation theory and algebraic geometry.
Orthogonality Relations: Orthogonality relations refer to the mathematical conditions that express how certain functions or representations are mutually perpendicular in a specific inner product space. In the context of character theory, these relations demonstrate how characters of irreducible representations of a finite group behave, particularly highlighting that distinct irreducible characters are orthogonal to each other under a certain inner product defined by group elements.
Philip Hall: Philip Hall was a British mathematician known for his significant contributions to group theory and combinatorics, particularly in the study of finite groups. His work introduced what is now referred to as Hall's Marriage Theorem, which provides conditions for the existence of a perfect matching in bipartite graphs, a crucial concept in character theory as it relates to representations of groups and their characters.
Principal character: A principal character is a specific type of character associated with a representation of a finite group that is derived from its irreducible representations. It serves as a fundamental building block in understanding the structure of characters in character theory, providing insight into the group's symmetry and behavior through the use of linear algebraic methods. The principal character corresponds to the trivial representation, where every group element is represented by the number one, highlighting the foundational nature of this concept in analyzing more complex characters.
Representation theory: Representation theory is a branch of mathematics that studies how algebraic structures, like groups and algebras, can be represented through linear transformations of vector spaces. This concept provides a way to connect abstract algebraic objects with more concrete linear algebra techniques, making it easier to analyze and understand their properties and behaviors.
William Burnside: William Burnside was a mathematician known for his contributions to group theory, particularly in character theory and representation theory. He formulated Burnside's lemma, which provides a way to count the number of distinct objects under group actions, thereby linking group theory to combinatorial counting problems.
χ (chi): In the context of character theory of finite groups, χ (chi) is a complex-valued function that assigns to each group element a scalar representing the trace of the corresponding group representation. It provides deep insights into the structure of a group by allowing us to study its representations through linear transformations. Characters are crucial for analyzing the irreducible representations of a group and play a significant role in counting conjugacy classes and understanding the character table.