Let $V$ be the vector space of all $n\times n$ real matrices.
Let us fix a matrix $A\in V$.
Define a map $T: V\to V$ by
\[ T(X)=AX-XA\]
for each $X\in V$.
(a) Prove that $T:V\to V$ is a linear transformation.
(b) Let $B$ be a basis of $V$. Let $P$ be the matrix representation of $T$ with respect to $B$. Find the determinant of $P$.
Add to solve later Let $f:G\to G’$ be a group homomorphism. We say that $f$ is monic whenever we have $fg_1=fg_2$, where $g_1:K\to G$ and $g_2:K \to G$ are group homomorphisms for some group $K$, we have $g_1=g_2$.
Then prove that a group homomorphism $f: G \to G’$ is injective if and only if it is monic.
Add to solve later Let $G$ be a finite group of order $n$ and let $m$ be an integer that is relatively prime to $n=|G|$. Show that for any $a\in G$, there exists a unique element $b\in G$ such that
\[b^m=a.\]
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Add to solve later Let $G$ and $H$ be groups and let $f:G \to K$ be a group homomorphism. Prove that the homomorphism $f$ is injective if and only if the kernel is trivial, that is, $\ker(f)=\{e\}$, where $e$ is the identity element of $G$.
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