Alphabeta Math
CorollaryStatement: Literature-sourcedProof: AI-generatedprecheck passaudited 2026-08-11
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GL⁡n(F) is a group under matrix multiplication, including the trivial group GL⁡0(F)

Statement

For every field F and natural n, GL⁡n(F) is a group under matrix multiplication. For n=0, it is the trivial group containing the unique empty matrix.

Facts & Assumptions

Given: A field F and a natural n.

[L1]

GL⁡n(F) is the set of invertible matrices, equivalently the units of Mn(F) (Invertible matrices and the general linear group GL⁡n(F)).

[L2]

The units of a ring contain the identity and are closed under multiplication and inversion, and they form a group (The units of a ring are the invertible elements of its multiplicative monoid, and R× is a group under multiplication; 0∈R× only in the zero ring).

Proof

technique · direct
1.1givenL1

By [L1], GL⁡n(F) is exactly the unit set of the ring Mn(F).

2.1step 1.1L1L2

Applying [L2] gives closure, associativity inherited from the ring, identity In, and inverse A−1 for every element, so GL⁡n(F) is a group.

3.1step 2.1L1L2∎

If n=0, M0(F) has one element, the empty matrix I0, which is its own inverse; hence its unit group is the trivial group.

Depends on

Dependency tree · 2 levels

Sources