# If Squares of Elements in a Group Lie in a Subgroup, then It is a Normal Subgroup

## Problem 469

Let $H$ be a subgroup of a group $G$.
Suppose that for each element $x\in G$, we have $x^2\in H$.

Then prove that $H$ is a normal subgroup of $G$.

(Purdue University, Abstract Algebra Qualifying Exam)

## Proof.

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 $G$.

### More from my site

• 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) […]
• 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 […]
• 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 […] • 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 […]
• 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 […]

#### You may also like...

This site uses Akismet to reduce spam. Learn how your comment data is processed.

##### Example of Two Groups and a Subgroup of the Direct Product that is Not of the Form of Direct Product

Give an example of two groups $G$ and $H$ and a subgroup $K$ of the direct product $G\times H$ such...

Close