Linear group in the context of Invertible matrix

⭐ Core Definition: Linear group

In mathematics, a matrix group is a group G consisting of invertible matrices over a specified field K, with the operation of matrix multiplication. A linear group is a group that is isomorphic to a matrix group (that is, admitting a faithful, finite-dimensional representation over K).

Any finite group is linear, because it can be realized by permutation matrices using Cayley's theorem. Among infinite groups, linear groups form an interesting and tractable class. Examples of groups that are not linear include groups which are "too big" (for example, the group of permutations of an infinite set), or which exhibit some pathological behavior (for example, finitely generated infinite torsion groups).

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⭐ Core Definition: Linear group
Linear group in the context of Invariant theory
Linear group in the context of Tits alternative

Linear group in the context of Invariant theory

Invariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties, such as vector spaces, from the point of view of their effect on functions. Classically, the theory dealt with the question of explicit description of polynomial functions that do not change (are invariant) under the transformations from a given linear group. For example, if we consider the action of the special linear group SL_n on the space of n by n matrices by left multiplication, then the determinant is an invariant of this action because the determinant of A X equals the determinant of X, when A is in SL_n.

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Linear group in the context of Tits alternative

In mathematics, the Tits alternative, named after Jacques Tits, is an important theorem about the structure of finitely generated linear groups.

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Linear group in the context of Invertible matrix

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⭐ Core Definition: Linear group

Linear group in the context of Invariant theory

Linear group in the context of Tits alternative