GL2m

GL2m, written GL(2, m) or GL2 for short in some texts, denotes a general linear group of degree 2 in two common interpretations. It can refer to the general linear group of 2×2 invertible matrices over a finite field with q elements, or to the group of 2×2 invertible matrices over the ring of integers modulo m.

In the finite field interpretation, GL(2, q) consists of all 2×2 matrices with entries in the finite field GF(q) and nonzero determinant. The action is on the two-dimensional vector space GF(q)^2 by matrix multiplication. The group is non-abelian for q ≥ 2 and has order |GL(2, q)| = (q^2 − 1)(q^2 − q) = q(q − 1)^2(q + 1). The projective general linear group PGL(2, q) is obtained by quotienting out the scalar matrices and has important applications in geometry, number theory, and representation theory.

In the ring interpretation, GL(2, Z/mZ) denotes the group of 2×2 matrices over the ring Z/mZ (integers modulo m) whose determinant is a unit modulo m, i.e., invertible modulo m. This interpretation is used when m is composite. Via the Chinese remainder theorem, GL(2, Z/mZ) ≅ ∏ GL(2, Z/p_i^{k_i}Z) for m factorized as ∏ p_i^{k_i}. For prime m = p, the order matches the finite-field case, |GL(2, p)| = (p^2 − 1)(p^2 − p). For higher powers p^k, the size depends on p and k in a way that can be determined from known formulas for GL(2, Z/p^kZ).

GL(2) groups appear in various areas of algebra, geometry, coding theory, and cryptography, often as building blocks in the study of linear transformations and group actions.