To show that $H$ is a normal subgroup of $G$, we prove that
\[ghg^{-1}\in H\]
for any $g\in G$ and $h\in H$.

For any $g\in G$ and $h\in H$ we have
\begin{align*}
&ghg^{-1}\\
&=g^2g^{-1}hg^{-1} &&\text{since $g=g^2g^{-1}$}\\
&=g^2g^{-1}hg^{-1}hh^{-1} &&\text{since $e=hh^{-1}$}\\
&=g^2(g^{-1}h)^2h^{-1}. \tag{*}
\end{align*}

It follows from the assumption that the elements $g^2$ and $(g^{-1}h)^2$ are in $H$.
Since $h\in H$, the inverse $h^{-1}$ is also in $H$.
Thus the expression in (*) is the product of elements in $H$, hence it is in $H$.

Thus, we have proved that $ghg^{-1}\in H$ for all $g\in G$, $h\in H$.
Therefore, the subgroup $H$ is a normal subgroup in $G$.

A Subgroup of Index a Prime $p$ of a Group of Order $p^n$ is Normal
Let $G$ be a finite group of order $p^n$, where $p$ is a prime number and $n$ is a positive integer.
Suppose that $H$ is a subgroup of $G$ with index $[G:P]=p$.
Then prove that $H$ is a normal subgroup of $G$.
(Michigan State University, Abstract Algebra Qualifying […]

Every Finite Group Having More than Two Elements Has a Nontrivial Automorphism
Prove that every finite group having more than two elements has a nontrivial automorphism.
(Michigan State University, Abstract Algebra Qualifying Exam)
Proof.
Let $G$ be a finite group and $|G|> 2$.
Case When $G$ is a Non-Abelian Group
Let us first […]

Simple Commutative Relation on Matrices
Let $A$ and $B$ are $n \times n$ matrices with real entries.
Assume that $A+B$ is invertible. Then show that
\[A(A+B)^{-1}B=B(A+B)^{-1}A.\]
(University of California, Berkeley qualifying exam)
Proof.
Let $P=A+B$. Then $B=P-A$.
Using these, we express the given […]

True or False Quiz About a System of Linear Equations
(Purdue University Linear Algebra Exam)
Which of the following statements are true?
(a) A linear system of four equations in three unknowns is always inconsistent.
(b) A linear system with fewer equations than unknowns must have infinitely many solutions.
(c) […]

Given a Spanning Set of the Null Space of a Matrix, Find the Rank
Let $A$ be a real $7\times 3$ matrix such that its null space is spanned by the vectors
\[\begin{bmatrix}
1 \\
2 \\
0
\end{bmatrix}, \begin{bmatrix}
2 \\
1 \\
0
\end{bmatrix}, \text{ and } \begin{bmatrix}
1 \\
-1 \\
0
[…]

A Simple Abelian Group if and only if the Order is a Prime Number
Let $G$ be a group. (Do not assume that $G$ is a finite group.)
Prove that $G$ is a simple abelian group if and only if the order of $G$ is a prime number.
Definition.
A group $G$ is called simple if $G$ is a nontrivial group and the only normal subgroups of $G$ is […]

A Matrix Equation of a Symmetric Matrix and the Limit of its Solution
Let $A$ be a real symmetric $n\times n$ matrix with $0$ as a simple eigenvalue (that is, the algebraic multiplicity of the eigenvalue $0$ is $1$), and let us fix a vector $\mathbf{v}\in \R^n$.
(a) Prove that for sufficiently small positive real $\epsilon$, the equation […]

Group of Order 18 is Solvable
Let $G$ be a finite group of order $18$.
Show that the group $G$ is solvable.
Definition
Recall that a group $G$ is said to be solvable if $G$ has a subnormal series
\[\{e\}=G_0 \triangleleft G_1 \triangleleft G_2 \triangleleft \cdots \triangleleft G_n=G\]
such […]