## Finite Group and Subgroup Criteria

## Problem 160

Let $G$ be a finite group and let $H$ be a subset of $G$ such that for any $a,b \in H$, $ab\in H$.

Then show that $H$ is a subgroup of $G$.

Read solution

Let $G$ be a finite group and let $H$ be a subset of $G$ such that for any $a,b \in H$, $ab\in H$.

Then show that $H$ is a subgroup of $G$.

Read solution

Let $T: \R^2 \to \R^2$ be a linear transformation.

Let

\[

\mathbf{u}=\begin{bmatrix}

1 \\

2

\end{bmatrix}, \mathbf{v}=\begin{bmatrix}

3 \\

5

\end{bmatrix}\]
be 2-dimensional vectors.

Suppose that

\begin{align*}

T(\mathbf{u})&=T\left( \begin{bmatrix}

1 \\

2

\end{bmatrix} \right)=\begin{bmatrix}

-3 \\

5

\end{bmatrix},\\

T(\mathbf{v})&=T\left(\begin{bmatrix}

3 \\

5

\end{bmatrix}\right)=\begin{bmatrix}

7 \\

1

\end{bmatrix}.

\end{align*}

Let $\mathbf{w}=\begin{bmatrix}

x \\

y

\end{bmatrix}\in \R^2$.

Find the formula for $T(\mathbf{w})$ in terms of $x$ and $y$.

Let $C[3, 10]$ be the vector space consisting of all continuous functions defined on the interval $[3, 10]$. Consider the set

\[S=\{ \sqrt{x}, x^2 \}\]
in $C[3,10]$.

Show that the set $S$ is linearly independent in $C[3,10]$.

Add to solve laterLet $P_2$ be the vector space of all polynomials of degree two or less.

Consider the subset in $P_2$

\[Q=\{ p_1(x), p_2(x), p_3(x), p_4(x)\},\]
where

\begin{align*}

&p_1(x)=x^2+2x+1, &p_2(x)=2x^2+3x+1, \\

&p_3(x)=2x^2, &p_4(x)=2x^2+x+1.

\end{align*}

**(a)** Use the basis $B=\{1, x, x^2\}$ of $P_2$, give the coordinate vectors of the vectors in $Q$.

**(b)** Find a basis of the span $\Span(Q)$ consisting of vectors in $Q$.

**(c)** For each vector in $Q$ which is not a basis vector you obtained in (b), express the vector as a linear combination of basis vectors.

Let $T: \R^3 \to \R^2$ be a linear transformation such that

\[T(\mathbf{e}_1)=\begin{bmatrix}

1 \\

4

\end{bmatrix}, T(\mathbf{e}_2)=\begin{bmatrix}

2 \\

5

\end{bmatrix}, T(\mathbf{e}_3)=\begin{bmatrix}

3 \\

6

\end{bmatrix},\]
where

\[\mathbf{e}_1=\begin{bmatrix}

1 \\

0 \\

0

\end{bmatrix}, \mathbf{e}_2=\begin{bmatrix}

0 \\

1 \\

0

\end{bmatrix}, \mathbf{e}_3=\begin{bmatrix}

0 \\

0 \\

1

\end{bmatrix}\]
are the standard unit basis vectors of $\R^3$.

For any vector $\mathbf{x}=\begin{bmatrix}

x_1 \\

x_2 \\

x_3

\end{bmatrix}\in \R^3$, find a formula for $T(\mathbf{x})$.

Let $A$ be an $m \times n$ matrix.

Let $\calN(A)$ be the null space of $A$. Suppose that $\mathbf{u} \in \calN(A)$ and $\mathbf{v} \in \calN(A)$.

Let $\mathbf{w}=3\mathbf{u}-5\mathbf{v}$.

Then find $A\mathbf{w}$.

Read solution

Define the map $T:\R^2 \to \R^3$ by $T \left ( \begin{bmatrix}

x_1 \\

x_2

\end{bmatrix}\right )=\begin{bmatrix}

x_1-x_2 \\

x_1+x_2 \\

x_2

\end{bmatrix}$.

**(a) **Show that $T$ is a linear transformation.

**(b)** Find a matrix $A$ such that $T(\mathbf{x})=A\mathbf{x}$ for each $\mathbf{x} \in \R^2$.

**(c)** Describe the null space (kernel) and the range of $T$ and give the rank and the nullity of $T$.

Let $P_3$ be the vector space over $\R$ of all degree three or less polynomial with real number coefficient.

Let $W$ be the following subset of $P_3$.

\[W=\{p(x) \in P_3 \mid p'(-1)=0 \text{ and } p^{\prime\prime}(1)=0\}.\]
Here $p'(x)$ is the first derivative of $p(x)$ and $p^{\prime\prime}(x)$ is the second derivative of $p(x)$.

Show that $W$ is a subspace of $P_3$ and find a basis for $W$.

Add to solve later Let $V$ be the vector space of all $2\times 2$ matrices, and let the subset $S$ of $V$ be defined by $S=\{A_1, A_2, A_3, A_4\}$, where

\begin{align*}

A_1=\begin{bmatrix}

1 & 2 \\

-1 & 3

\end{bmatrix}, \quad

A_2=\begin{bmatrix}

0 & -1 \\

1 & 4

\end{bmatrix}, \quad

A_3=\begin{bmatrix}

-1 & 0 \\

1 & -10

\end{bmatrix}, \quad

A_4=\begin{bmatrix}

3 & 7 \\

-2 & 6

\end{bmatrix}.

\end{align*}

Find a basis of the span $\Span(S)$ consisting of vectors in $S$ and find the dimension of $\Span(S)$.

Let $B=\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\}$ be a basis for a vector space $V$ over a scalar field $K$. Then show that any vector $\mathbf{v}\in V$ can be written uniquely as

\[\mathbf{v}=c_1\mathbf{v}_1+c_2\mathbf{v}_2+c_3\mathbf{v}_3,\]
where $c_1, c_2, c_3$ are scalars.

Read solution

Show that the set

\[S=\{1, 1-x, 3+4x+x^2\}\]
is a basis of the vector space $P_2$ of all polynomials of degree $2$ or less.

Let $G$ be a non-abelian simple group. Let $D(G)=[G,G]$ be the commutator subgroup of $G$. Show that $G=D(G)$.

Read solution

Let $K, N$ be normal subgroups of a group $G$. Suppose that the quotient groups $G/K$ and $G/N$ are both abelian groups.

Then show that the group

\[G/(K \cap N)\]
is also an abelian group.

Read solution

Let $G$ be a group and let $D(G)=[G,G]$ be the commutator subgroup of $G$.

Let $N$ be a subgroup of $G$.

Prove that the subgroup $N$ is normal in $G$ and $G/N$ is an abelian group if and only if $N \supset D(G)$.

Let $A$ be an $n \times n$ nilpotent matrix, that is, $A^m=O$ for some positive integer $m$, where $O$ is the $n \times n$ zero matrix.

Prove that $A$ is a singular matrix and also prove that $I-A, I+A$ are both nonsingular matrices, where $I$ is the $n\times n$ identity matrix.

Add to solve laterLet $G$ 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

\[b^m=a.\]

Read solution

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 $G$.

Read solution

Let $V$ be the vector space over $\R$ consisting of all $n\times n$ real matrices for some fixed integer $n$. Prove or disprove that the following subsets of $V$ are subspaces of $V$.

**(a) **The set $S$ consisting of all $n\times n$ symmetric matrices.

**(b)** The set $T$ consisting of all $n \times n$ skew-symmetric matrices.

**(c)** The set $U$ consisting of all $n\times n$ nonsingular matrices.

Let $T:\R^2 \to \R^3$ be a linear transformation such that $T(\mathbf{e}_1)=\mathbf{u}_1$ and $T(\mathbf{e}_2)=\mathbf{u}_2$, where $\mathbf{e}_1=\begin{bmatrix}

1 \\

0

\end{bmatrix}, \mathbf{e}_2=\begin{bmatrix}

0 \\

1

\end{bmatrix}$ are unit vectors of $\R^2$ and

\[\mathbf{u}_1= \begin{bmatrix}

-1 \\

0 \\

1

\end{bmatrix}, \quad \mathbf{u}_2=\begin{bmatrix}

2 \\

1 \\

0

\end{bmatrix}.\]
Then find $T\left(\begin{bmatrix}

3 \\

-2

\end{bmatrix}\right)$.

Let $V$ be a vector space over a field $K$. Let $\mathbf{u}_1, \mathbf{u}_2, \dots, \mathbf{u}_n$ be linearly independent vectors in $V$. Let $U$ be the subspace of $V$ spanned by these vectors, that is, $U=\Span \{\mathbf{u}_1, \mathbf{u}_2, \dots, \mathbf{u}_n\}$.

Let $\mathbf{u}_{n+1}\in V$. Show that $\mathbf{u}_1, \mathbf{u}_2, \dots, \mathbf{u}_n, \mathbf{u}_{n+1}$ are linearly independent if and only if $\mathbf{u}_{n+1} \not \in U$.