PGLn1whether

PGLn1whether is a theoretical construct in abstract algebra and computational group theory that denotes a decision problem related to embeddings into projective general linear groups. The name combines PGL(n−1) with the English word whether, signaling a question: does a given group admit an embedding into a projective general linear group of dimension n−1 over a field F? Note that PGLn1whether is not a standard term in established literature; it is used here as a hypothetical concept to illustrate how embedding questions might be formalized.

Definition and inputs. In its common formulation, the input consists of a finitely presented group G (or a generating set for a matrix group in GL(k, F)), together with fixed integers n and a field F. The decision problem asks whether there exists an injective homomorphism from G into PGL(n−1, F) up to projective equivalence. Variants may fix n or allow it to vary, or may specify different representations of G (e.g., as a permutation or matrix group).

Theoretical context. PGLn1whether intersects with representation theory and the study of projective representations. Its core is the question of lifting a projective representation to a linear one and understanding when such an embedding is possible. The problem is sensitive to the choice of field F and to the dimensions involved, and it connects to classical concepts such as Schur multipliers and central extensions.

Complexity and examples. There are no general efficient algorithms for all cases; the problem is typically challenging and can be computationally hard in many settings. In simple instances, trivial groups or small cyclic groups may embed into PGL(n−1, F) for suitable n and F.

Applications and related topics. PGLn1whether serves as a conceptual bridge to questions about symmetries in geometry, group representations in computer algebra, and theoretical aspects of embedding problems. See also PGL, GL, projective representations, and embedding problems. This article presents PGLn1whether as a fictional, illustrative term rather than a term with established usage.