CorollaryStatement: Literature-sourcedProof: AI-generatedprecheck passaudited 2026-08-11

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

${GL}_{n} (F)$ 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 $M_{n} (F)$ (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 $R^{\times}$ is a group under multiplication; $0 \in R^{\times}$ only in the zero ring).

Proof

technique · direct

1.1

By [L1], ${GL}_{n} (F)$ is exactly the unit set of the ring $M_{n} (F)$ .

givenL1

2.1

Applying [L2] gives closure, associativity inherited from the ring, identity $I_{n}$ , and inverse $A^{- 1}$ for every element, so ${GL}_{n} (F)$ is a group.

step 1.1L1L2

3.1

If $n = 0$ , $M_{0} (F)$ has one element, the empty matrix $I_{0}$ , which is its own inverse; hence its unit group is the trivial group.

step 2.1L1L2∎

Depends on

Used by

Nothing in the library uses this result yet.

Cited to discharge well-definedness by Invertible matrices and the general linear group GLₙ(F).

Dependency tree · two levels

13 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

S. Axler, Linear Algebra Done Right, 4th ed., §3D (standard reference, not scraped)

GL⁡n(F)GLn​(F) is a group under matrix multiplication, including the trivial group GL⁡0(F)GL0​(F)