Group Generated by Commutators of Two Normal Subgroups is a Normal Subgroup
Let $G$ be a group and $H$ and $K$ be subgroups of $G$.
For $h \in H$, and $k \in K$, we define the commutator $[h, k]:=hkh^{-1}k^{-1}$.
Let $[H,K]$ be a subgroup of $G$ generated by all such commutators.
Show that if $H$ and $K$ are normal subgroups of $G$, then the subgroup […]

Every Ideal of the Direct Product of Rings is the Direct Product of Ideals
Let $R$ and $S$ be rings with $1\neq 0$.
Prove that every ideal of the direct product $R\times S$ is of the form $I\times J$, where $I$ is an ideal of $R$, and $J$ is an ideal of $S$.
Proof.
Let $K$ be an ideal of the direct product $R\times […]

Find a Polynomial Satisfying the Given Conditions on Derivatives
Find a cubic polynomial
\[p(x)=a+bx+cx^2+dx^3\]
such that $p(1)=1, p'(1)=5, p(-1)=3$, and $ p'(-1)=1$.
Solution.
By differentiating $p(x)$, we obtain
\[p'(x)=b+2cx+3dx^2.\]
Thus the given conditions are
[…]

Isomorphism Criterion of Semidirect Product of Groups
Let $A$, $B$ be groups. Let $\phi:B \to \Aut(A)$ be a group homomorphism.
The semidirect product $A \rtimes_{\phi} B$ with respect to $\phi$ is a group whose underlying set is $A \times B$ with group operation
\[(a_1, b_1)\cdot (a_2, b_2)=(a_1\phi(b_1)(a_2), b_1b_2),\]
where $a_i […]

A Recursive Relationship for a Power of a Matrix
Suppose that the $2 \times 2$ matrix $A$ has eigenvalues $4$ and $-2$. For each integer $n \geq 1$, there are real numbers $b_n , c_n$ which satisfy the relation
\[ A^{n} = b_n A + c_n I , \]
where $I$ is the identity matrix.
Find $b_n$ and $c_n$ for $2 \leq n \leq 5$, and […]