General linear group over a field

This article defines a natural number-parametrized system of algebraic matrix groups. In other words, for every field and every natural number, we get a matrix group defined by a system of algebraic equations. The definition may also generalize to arbitrary commutative unital rings, though the default usage of the term is over fields.
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This article is about the unit group (group of invertible elements) in the following ring/monoid: matrix ring

This article is about a general term. A list of important particular cases (instances) is available at Category:General linear groups

Definition

In terms of dimension

Let $n$ be a natural number and $k$ a field. The general linear group of order $n$ over $k$, denoted $GL(n,k)$, is defined in the following equivalent ways:

• $GL(n,k)$ is the group of all invertible $k$-linear maps from the vector space $k^n$ to itself, under composition. In other words, it is the group of automorphisms of $k^n$ as a $k$-vector space.
• $GL(n,k)$ is the group of all invertible $n\times n$ matrices with entries over $k$

In terms of vector spaces

Let $V$ be a $k$-vector space (which may be finite or infinite-dimensional). The general linear group over $V$, denoted $GL(V)$, is the group of all vector space automorphisms from $V$ to itself.

Note that when $V=k^n$, this reduces to the definition $Gl(n,k)$. Further, since $GL(V)\cong GL(W)$ for $V\cong W$, and since any two vectro spaces of the same dimension are isomorphic, the $GL(n,k)$s cover all general linear groups corresponding to finite-dimensional vector spaces.

As a map

As a functor from fields to groups

If we fix $n$, we can think of $G{L}_n$ as a functor from the category of fields to the category of groups.

As an IAPS

Further information: GL IAPS

For a fixed field $k$, the general linear groups $GL(n,k)$ form an IAPS of groups parametrized by $n$. In other words, we naturally have concatenation maps:

${\mathrm{\Phi }}_{m,n}:GL(m,k)\times GL(n,k)\to GL(m+n,k)$

This map takes a matrix $A$ of order $m$ and a matrix $B$ of order $n$ and putputs the block diagonal matrix with blocks $A$ and $B$.

As a functor from fields to IAPSes

If we fix neither $n$ nor $k$, we can view $GL$ as a functor from fields to the category of IAPSes of groups.

Subgroups

The general linear group has a number of important subgroups. Some of them are listed below:

• Orthogonal group is the subgroup comprising those matrices $A$ such that $A{A}^t$ is the identity matrix. Further information: Orthogonal group in general linear group
• Special linear group is the subgroup comprising those matrices with determinant 1. Further information: Special linear group in general linear group
• Symplectic group is the subgroup comprising matrices $A$ such that $AP{A}^t=P$ where $P$ is a certain fixed matrix. Further information: Symplectic group in general linear group
• Special orthogonal group is the intersection of the orthogonal group and the special linear group
• Orthogonal similitude group is the group of matrices $A$ such that $A{A}^t$ is a scalar matrix
• Symmetric group which is the group of permutation matrices, embedded in the general linear group

All of these, except the orthogonal similtude group, form sub-IAPSes.

Quotients

Inner automorphism group

The inner automorphism group of this group, viz the quotient group by its center, is abstractly isomorphic to: projective general linear group

Supergroups

Some important groups in which the general linear group is contained:

• General affine group which is the semidirect product of the vector space (as an Abelian group) and the general linear group acting on it
• Skew-linear group which is the semidirect product of the general linear group with the transpose-inverse map

Extensions

Groups having the general linear group as a quotient?