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Magazine Chapter 3: Problem 12
(i) Let \(k\) be a field. If \(G=G L(n, k)\) and \(T\) is the subgroup of \(G\) of all diagonal matrices, then \(N_{G}(T)\) consists of all the monomial matrices over \(k\). (ii) Prove that \(N_{G}(T) / T \cong S_{n}\) -
Short Answer
Expert verified
N_G(T) consists of all monomial matrices in GL(n,k). N_G(T) / T is isomorphic to S_n.
Step by step solution
01
- Define Groups and Subgroups
Define the group G and subgroup T: Let G = GL(n, k) where k is a field, representing the general linear group of all n x n invertible matrices over k. Let T be the subgroup of G consisting of all diagonal matrices.
02
- Normalizer Definition
Define the normalizer of T in G, denoted by N_G(T). This is the set of all elements in G that conjugate T into itself: N_G(T) = {g 鈭� G | gTg^{-1} 鈯� T}.
03
- Conjugation of Diagonal Matrices
Consider a matrix D in T. For a matrix g in G to be in N_G(T), conjugation gDg^{-1} must remain a diagonal matrix for all diagonal matrices D in T. This indicates that g must be a monomial matrix (a matrix with exactly one non-zero entry from each row and each column).
04
- Monomial Matrix Structure
Conclude that matrices in N_G(T) are monomial matrices. These are the matrices that can be written as a product of a permutation matrix and a diagonal matrix.
05
- Quotient Group and Isomorphism
Consider the quotient group N_G(T) / T. Each coset in N_G(T) / T can be uniquely associated with a permutation of the n indices. This means that each coset corresponds uniquely to an element of the symmetric group S_n.
06
- Prove the Isomorphism
Finally, show that the map from N_G(T) / T to S_n is indeed a homomorphism and it is bijective, hence proving that N_G(T) / T is isomorphic to S_n.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
General Linear Group
The General Linear Group, denoted as GL(n, k), is a fundamental concept in Group Theory. It consists of all invertible n x n matrices over a given field k. An invertible matrix is one that has an inverse, meaning there exists another matrix that, when multiplied, results in the identity matrix.
The identity matrix is a special diagonal matrix with ones on the diagonal and zeros elsewhere.
Key characteristics of GL(n, k) include:
The identity matrix is a special diagonal matrix with ones on the diagonal and zeros elsewhere.
Key characteristics of GL(n, k) include:
- Closed under matrix multiplication - If A and B are in GL(n, k), then AB is also in GL(n, k).
- Existence of the identity element - There is an identity matrix I such that AI = IA = A for any A in GL(n, k).
- Existence of inverses - For any matrix A in GL(n, k), there exists an inverse matrix A^-1 such that AA^-1 = A^-1A = I.
Normalizer
The Normalizer is a concept used to study how subgroups behave within larger groups.
Specifically, the normalizer N_G(T) of a subgroup T in a group G consists of all elements in G that, when conjugated, keep T within itself.
In mathematical terms, N_G(T) = {g 鈭� G | gTg^{-1} 鈯� T}.
This means for g to be in N_G(T), conjugating any element of T by g must result in another element in T.
Here are some essential points:
Specifically, the normalizer N_G(T) of a subgroup T in a group G consists of all elements in G that, when conjugated, keep T within itself.
In mathematical terms, N_G(T) = {g 鈭� G | gTg^{-1} 鈯� T}.
This means for g to be in N_G(T), conjugating any element of T by g must result in another element in T.
Here are some essential points:
- Identity element is always in the normalizer as it does nothing when conjugating.
- Normalizers help in studying the symmetry and structure of groups.
Monomial Matrix
A Monomial Matrix is an n x n matrix that has exactly one non-zero entry in each row and each column. These non-zero entries can be any number from the field k.
Monomial matrices can be represented as a product of a diagonal matrix and a permutation matrix.
This can be broken down as follows:
They maintain diagonal structure when conjugated, which is needed for normalizing T.
Monomial matrices can be represented as a product of a diagonal matrix and a permutation matrix.
This can be broken down as follows:
- Permutation Matrix: A matrix where exactly one entry in each row and column is 1, and all other entries are 0.
- Diagonal Matrix: A matrix with non-zero entries only on the main diagonal and zeros elsewhere.
They maintain diagonal structure when conjugated, which is needed for normalizing T.
Permutation Matrix
A Permutation Matrix is a special type of matrix used frequently in algebra and combinatorics.
It is an n x n matrix where each row and column contains exactly one entry of 1, with all other entries being 0.
Permutation matrices correspond to permutations of the set {1, 2, ... , n}.
Here's why they are important:
It is an n x n matrix where each row and column contains exactly one entry of 1, with all other entries being 0.
Permutation matrices correspond to permutations of the set {1, 2, ... , n}.
Here's why they are important:
- They represent reordering or shuffling of elements.
- Multiplying a matrix by a permutation matrix permutes the rows or columns of the original matrix.
- They are special cases of monomial matrices where the non-zero entries are specifically ones.
Symmetric Group
The Symmetric Group S_n is a fundamental structure in algebra that contains all possible permutations of n elements.
It consists of all bijective functions from a set of n elements to itself.
Key facts about S_n:
This isomorphism signifies that each coset corresponds to a unique permutation, further indicating the rich interplay between matrices and permutation groups in group theory.
It consists of all bijective functions from a set of n elements to itself.
Key facts about S_n:
- It has n! (n factorial) elements.
- Each element corresponds to a unique way of permuting n objects.
- It plays a crucial role in understanding the symmetries of objects.
This isomorphism signifies that each coset corresponds to a unique permutation, further indicating the rich interplay between matrices and permutation groups in group theory.
