Problem 9 - Algebra

Question

Let \(A\) be a commutative subgroup of a finite group \(G .\) Show that every irreducible representation of \(G\) over \(\mathbf{C}\) has dimension \(\leqq(G: A)\).

Step-by-Step Solution

Answer

In summary, we can show that every irreducible representation of $G$ over $\mathbf{C}$ has dimension at most $(G:A)$ by applying Schur's Lemma, which implies that elements in the commutative subgroup $A$ are represented by scalar matrices. Since the order of $A$ is at most the dimension of any irreducible representation, and the order of the subgroup $A$ is $(G:A)$ times smaller than the order of $G$, it follows that the dimension of any irreducible representation of $G$ is at most $(G:A)$.

Step 1: Understanding of Irreducible Representation

An irreducible representation is a representation that is not equivalent to a direct sum of two or more nontrivial (i.e., dimension greater than 1) representations. In other words, there are no nontrivial subspaces which are invariant under all the transformations of the group G.

Step 2: Understanding of Commutative Subgroup

The subgroup A is commutative which means that for any two elements a, b in A, ab = ba. This property will be utilized later.

Step 3: Application of Schur's Lemma

By applying Schur's Lemma, for the commutative subgroup A, in any irreducible representation, the elements of A are represented by scalar matrices, because in any irreducible representation any operator commuting with all operators of the representation must be a scalar multiple of the identity.

Step 4: Observing Dimensions

From Step 3, it can be concluded that if a representation is irreducible, then, in particular, the identity of the group (usually denoted as e) also acts as a scalar operator. Therefore this scalar must be 1 since the identity must be represented by the identity matrix. Hence, the order of A is at most the dimension of the representation, because all the scalar operators representing elements of A should be different, as each group element corresponds to a unique matrix in the representation.

Step 5: Concluding the Result

Given G is a finite group and A is a subgroup of G, we can conclude that the order of the subgroup A is (G:A) times smaller than the order of G. Thus, since the order of A is at most the dimension of any irreducible representation by Step 4, the irreducible representation of G has dimension at most (G : A). This completes the solution.

Key Concepts

Commutative Subgroup

A commutative subgroup, also known as an abelian subgroup, is a collection of elements within a group that follow the commutative property. This means if you have two elements, say \(a\) and \(b\), in the subgroup, then multiplying them in either order results in the same element, i.e., \(ab = ba\).
This characteristic is crucial because it simplifies many group operations, allowing more straightforward computations and implications in group theory. Commutative subgroups play a significant role when dealing with representations, especially because they are easier to analyze over other subgroup structures.
In the context of the exercise, the commutative nature of subgroup \(A\) indicates that its elements, when represented in any manner—such as through matrices, will abide by this simple permutation rule. This property helps derive further results using Schur's Lemma, which exploits this commutativity in analyzing group representations.

Schur's Lemma

Schur's Lemma is a powerful tool in representation theory. It states that if you have an irreducible representation of a group and a linear operator that commutes with every operation in this representation, then that operator must be a scalar multiple of the identity.
In simpler terms, for an irreducible representation, the elements that commute with the entire representation do not add any 'new' data—they simply scale what is already there. This result is instrumental when examining representations because it allows us to restrict and control the form of our representations.
In the exercise, Schur's Lemma tells us that within any irreducible representation of the commutative subgroup \(A\), the group's elements can only be represented by scalar matrices. The presence of scalar matrices ensures that the dimensions of representations stay within a limit, adhering to the subgroup's order, which is vital for proving the exercise's assertion about dimensions.

Finite Group

A finite group is a set with a limited, countable number of elements that together form a group under a particular operation. Finite groups are especially important in constraints and symmetry in mathematical systems.
The concept of the group order, which is the total number of elements, becomes pivotal in group representations. In our exercise, since \(G\) is finite, we can well-define its subgroup \(A\) and their relation through the index \((G:A)\).
The finite nature of \(G\) is what allows us to count and compare dimensions effectively. When handling finite groups, considerations like subgroup indices, direct products, and group actions become more manageable, providing a structured way to derive properties such as those found in irreducible representations.

Group Representation

Group representation is a very useful correspondence between group theory and linear algebra. It refers to a way of expressing group operations as matrices and vectors, thus allowing the powerful tools of linear algebra to analyze group properties.
In group representation theory, we focus on homomorphisms from a group \(G\) to the general linear group of matrices over a field \(F\). These representations express group elements as invertible matrices, where group operations translate into matrix multiplication.
For the exercise, analyzing the representation of \(G\) helps to determine how irreducible representations function. Since we're dealing with an irreducible representation of a finite group \(G\) over \(\mathbf{C}\), we aim to see how the relevant dimensions compare in the context of this correspondence, mainly how they compare to the index \((G : A)\). Irreducible representations simplify this by containing minimal dimensional information yet offer crucial insights into more complex representation behaviors.