General linear group over subspace is conjugacy-closed

This article gives the statement, and proof, of a particular subgroup in a group being conjugacy-closed: in other words, any two elements of the subgroup that are conjugate in the whole group, are also conjugate in the subgroup
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Statement

Suppose $V$ is a finite-dimensional vector space over a field $k$, written as a direct sum of subspaces $U$ and $W$. Then, consider the map:

$i:GL(U)\to GL(V)$

that sends a linear map on $U$ to a linear map on $V$ that behaves the same way on $U$ and is the identity on $W$. This is an injective homomorphism, hence we can identify $GL(U)$ with a subgroup of $GL(V)$.

Then, if $A,B\in GL(U)$ are such that the images $i(A)$ and $i(B)$ are conjugate in $GL(V)$, then $A,B$ are conjugate in $GL(U)$.

Related facts

Other related facts

• Brauer's permutation lemma
• General linear group over subfield is conjugacy-closed
• Symmetric group on finite or cofinite subset is conjugacy-closed