General linear group - Wikipedia

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Abstraction: Group of invertible n-by-n matrices under multiplication

Key points:

GL(n,F) is the set of all invertible n×n matrices over a field F with matrix multiplication; named because columns of invertible matrices are linearly independent
Special linear group SL(n) is the normal subgroup of determinant-1 matrices; GL(n)/SL(n) is isomorphic to the multiplicative group of the field
Over real numbers, GL(n,R) is a Lie group of dimension n² with two connected components (positive and negative determinant); maximal compact subgroup is O(n)
Over finite field GF(q), |GL(n,q)| = (q^n−1)(q^n−q)···(q^n−q^(n−1)); GL(3,2) has order 168, isomorphic to PSL(2,7) and automorphism group of the Fano plane
Infinite stable general linear group GL(∞) is the direct limit used in algebraic K-theory (defines K1); has topology described by Bott periodicity
Galois computed the order of GL(n,p) over prime fields in 1832

Connections: Group Theory · Linear Algebra · Lie Groups