Let us define operators phiA acting on the space Matn of n t

Let us define operators phi_A acting on the space Mat_n of n times n matrices by the formula phi_A (X) = A^2 X. Here A GL_n is a non-degenerate n times n matrix, X Mat_n. Find natural numbers n such that the operators phi_A specify an action of GL_n on Mat_n.

Solution

A matrix group, or linear group, is a group G whose elements are invertible n × n matrices over a field F. The general linear group GL(n,F) is the group consisting of all invertible n×n matrices over the field F. So a group G is a matrix group precisely when it is a subgroup of GL(n,F) for some natural number n and field F. If V is a vector space of dimension n over F, we denote by GL(V) the group of invertible linear transformations of V; thus GL(V) GL(n,F).

If every finitely generated subgroup of a group G is isomorphic to a linear group of degree n, then G is isomorphic to a linear group of degree n. The same assertion holds in any given characteristic.

Any free group is linear of degree 2 in every characteristic. More generally, a free product of linear groups is linear. A matrix group G GL(V) is said to be reducible if there is a G-invariant subspace U of V other than {0} and V, and is irreducible otherwise. A matrix group G GL(V) is said to be decomposable if V is the direct sum of two non-zero G-invariant subspaces, and is indecomposable otherwise. If V can be expressed as the direct sum of irreducible subspaces, then G is completely reducible. (Thus an irreducible group is completely reducible.) If the matrix group G GL(n,F) is irreducible regarded as a subgroup of GL(n,K) for any algebraic extension K of F, we say that G is absolutely irreducible.