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matrix group
In mathematics, a matrix group is a group ''G'' consisting of invertible matrices over some field ''K'', usually fixed in advance, with operations of matrix multiplication and inversion. More generally, one can consider ''n'' × ''n'' matrices over a commutative ring ''R''. (The size of the matrices is restricted to be finite, as any group can be represented as a group of infinite matrices over any field.) A linear group is an abstract group that is isomorphic to a matrix group over a field ''K'', in other words, 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 all "sufficiently large" groups; for example, the infinite symmetric group of permutations of an infinite set.
==Basic examples==
The set ''M''''R''(''n'',''n'') of ''n'' × ''n'' matrices over a commutative ring ''R'' is itself a ring under matrix addition and multiplication. The group of units of ''M''''R''(''n'',''n'') is called the general linear group of ''n'' × ''n'' matrices over the ring ''R'' and is denoted ''GL''''n''(''R'') or ''GL''(''n'',''R''). All matrix groups are subgroups of some general linear group.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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