## Any Finite Group Has a Composition Series

## Problem 122

Let $G$ be a finite group. Then show that $G$ has a composition series.

Add to solve laterof the day

Let $G$ be a finite group. Then show that $G$ has a composition series.

Add to solve later Let $A$ be an $m \times n$ real matrix. Then the **null space** $\calN(A)$ of $A$ is defined by

\[ \calN(A)=\{ \mathbf{x}\in \R^n \mid A\mathbf{x}=\mathbf{0}_m\}.\]
That is, the null space is the set of solutions to the homogeneous system $A\mathbf{x}=\mathbf{0}_m$.

Prove that the null space $\calN(A)$ is a subspace of the vector space $\R^n$.

(Note that the null space is also called the **kernel** of $A$.)

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Suppose that $\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_r$ are linearly dependent $n$-dimensional real vectors.

For any vector $\mathbf{v}_{r+1} \in \R^n$, determine whether the vectors $\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_r, \mathbf{v}_{r+1}$ are linearly independent or linearly dependent.

Add to solve laterLet $\mathbf{a}$ and $\mathbf{b}$ be fixed vectors in $\R^3$, and let $W$ be the subset of $\R^3$ defined by

\[W=\{\mathbf{x}\in \R^3 \mid \mathbf{a}^{\trans} \mathbf{x}=0 \text{ and } \mathbf{b}^{\trans} \mathbf{x}=0\}.\]

Prove that the subset $W$ is a subspace of $\R^3$.

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Let $G$ be a finite group of order $18$.

Show that the group $G$ is solvable.

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Let $G$ be a finite group and $P$ be a nontrivial Sylow subgroup of $G$.

Let $H$ be a subgroup of $G$ containing the normalizer $N_G(P)$ of $P$ in $G$.

Then show that $N_G(H)=H$.

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Let $G$ and $G’$ be groups and let $f:G \to G’$ be a group homomorphism.

If $H’$ is a normal subgroup of the group $G’$, then show that $H=f^{-1}(H’)$ is a normal subgroup of the group $G$.

Express the vector $\mathbf{b}=\begin{bmatrix}

2 \\

13 \\

6

\end{bmatrix}$ as a linear combination of the vectors

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

1 \\

5 \\

-1

\end{bmatrix},

\mathbf{v}_2=

\begin{bmatrix}

1 \\

2 \\

1

\end{bmatrix},

\mathbf{v}_3=

\begin{bmatrix}

1 \\

4 \\

3

\end{bmatrix}.\]

(*The Ohio State University, Linear Algebra Exam*)

Let

\[A=\begin{bmatrix}

-1 & 2 \\

0 & -1

\end{bmatrix} \text{ and } \mathbf{u}=\begin{bmatrix}

1\\

0

\end{bmatrix}.\]
Compute $A^{2017}\mathbf{u}$.

(*The Ohio State University, Linear Algebra Exam*)

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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 $a_i \in A, b_i \in B$ for $i=1, 2$.

Let $f: A \to A’$ and $g:B \to B’$ be group isomorphisms. Define $\phi’: B’\to \Aut(A’)$ by sending $b’ \in B’$ to $f\circ \phi(g^{-1}(b’))\circ f^{-1}$.

\[\require{AMScd}

\begin{CD}

B @>{\phi}>> \Aut(A)\\

@A{g^{-1}}AA @VV{\sigma_f}V \\

B’ @>{\phi’}>> \Aut(A’)

\end{CD}\]
Here $\sigma_f:\Aut(A) \to \Aut(A’)$ is defined by $ \alpha \in \Aut(A) \mapsto f\alpha f^{-1}\in \Aut(A’)$.

Then show that

\[A \rtimes_{\phi} B \cong A’ \rtimes_{\phi’} B’.\]

Let $G$ be a simple group and let $X$ be a finite set.

Suppose $G$ acts nontrivially on $X$. That is, there exist $g\in G$ and $x \in X$ such that $g\cdot x \neq x$.

Then show that $G$ is a finite group and the order of $G$ divides $|X|!$.

Let $A$ and $B$ be $n \times n$ real symmetric matrices. Prove the followings.

**(a)** The product $AB$ is symmetric if and only if $AB=BA$.

**(b)** If the product $AB$ is a diagonal matrix, then $AB=BA$.

Let $p \in \Z$ be a prime number.

Then describe the elements of the Galois group of the polynomial $x^p-2$.

Add to solve laterLet $X$ be a subset of a group $G$. Let $C_G(X)$ be the centralizer subgroup of $X$ in $G$.

For any $g \in G$, show that $gC_G(X)g^{-1}=C_G(gXg^{-1})$.

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Let $\F_p$ be the finite field of $p$ elements, where $p$ is a prime number.

Let $G_n=\GL_n(\F_p)$ be the group of $n\times n$ invertible matrices with entries in the field $\F_p$. As usual in linear algebra, we may regard the elements of $G_n$ as linear transformations on $\F_p^n$, the $n$-dimensional vector space over $\F_p$. Therefore, $G_n$ acts on $\F_p^n$.

Let $e_n \in \F_p^n$ be the vector $(1,0, \dots,0)$.

(The so-called first standard basis vector in $\F_p^n$.)

Find the size of the $G_n$-orbit of $e_n$, and show that $\Stab_{G_n}(e_n)$ has order $|G_{n-1}|\cdot p^{n-1}$.

Conclude by induction that

\[|G_n|=p^{n^2}\prod_{i=1}^{n} \left(1-\frac{1}{p^i} \right).\]

For what value(s) of $a$ does the system have nontrivial solutions?

\begin{align*}

&x_1+2x_2+x_3=0\\

&-x_1-x_2+x_3=0\\

& 3x_1+4x_2+ax_3=0.

\end{align*}

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Let $G$ be a finite group of odd order. Assume that $x \in G$ is not the identity element.

Show that $x$ is not conjugate to $x^{-1}$.

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Let $G$ be a finite group of order $n$ and suppose that $p$ is the smallest prime number dividing $n$.

Then prove that any subgroup of index $p$ is a normal subgroup of $G$.

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Test your understanding of basic properties of matrix operations.

There are **10 True or False Quiz Problems**.

These 10 problems are very common and essential.

So make sure to understand these and don’t lose a point if any of these is your exam problems.

(These are actual exam problems at the Ohio State University.)

You can take the quiz as many times as you like.

The solutions will be given after completing all the 10 problems.

Click the **View question** button to see the solutions.

Find the rank of the following real matrix.

\[ \begin{bmatrix}

a & 1 & 2 \\

1 &1 &1 \\

-1 & 1 & 1-a

\end{bmatrix},\]
where $a$ is a real number.

(*Kyoto University, Linear Algebra Exam*)

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