## Every Group of Order 24 Has a Normal Subgroup of Order 4 or 8

## Problem 568

Prove that every group of order $24$ has a normal subgroup of order $4$ or $8$.

Add to solve laterProve that every group of order $24$ has a normal subgroup of order $4$ or $8$.

Add to solve later Let $N$ be a normal subgroup of a group $G$.

Suppose that $G/N$ is an infinite cyclic group.

Then prove that for each positive integer $n$, there exists a normal subgroup $H$ of $G$ of index $n$.

Add to solve later Let

\[R=\left\{\, \begin{bmatrix}

a & b\\

0& a

\end{bmatrix} \quad \middle | \quad a, b\in \Q \,\right\}.\]
Then the usual matrix addition and multiplication make $R$ an ring.

Let

\[J=\left\{\, \begin{bmatrix}

0 & b\\

0& 0

\end{bmatrix} \quad \middle | \quad b \in \Q \,\right\}\]
be a subset of the ring $R$.

**(a)** Prove that the subset $J$ is an ideal of the ring $R$.

**(b)** Prove that the quotient ring $R/J$ is isomorphic to $\Q$.

Let $R$ be a commutative ring. Consider the polynomial ring $R[x,y]$ in two variables $x, y$.

Let $(x)$ be the principal ideal of $R[x,y]$ generated by $x$.

Prove that $R[x, y]/(x)$ is isomorphic to $R[y]$ as a ring.

Add to solve later Let $R$ be a commutative ring with $1$ and let $M$ be an $R$-module.

Prove that the $R$-module $M$ is irreducible if and only if $M$ is isomorphic to $R/I$, where $I$ is a maximal ideal of $R$, as an $R$-module.

Let $\calF[0, 2\pi]$ be the vector space of all real valued functions defined on the interval $[0, 2\pi]$.

Define the map $f:\R^2 \to \calF[0, 2\pi]$ by

\[\left(\, f\left(\, \begin{bmatrix}

\alpha \\

\beta

\end{bmatrix} \,\right) \,\right)(x):=\alpha \cos x + \beta \sin x.\]
We put

\[V:=\im f=\{\alpha \cos x + \beta \sin x \in \calF[0, 2\pi] \mid \alpha, \beta \in \R\}.\]

**(a)** Prove that the map $f$ is a linear transformation.

**(b)** Prove that the set $\{\cos x, \sin x\}$ is a basis of the vector space $V$.

**(c)** Prove that the kernel is trivial, that is, $\ker f=\{\mathbf{0}\}$.

(This yields an isomorphism of $\R^2$ and $V$.)

**(d)** Define a map $g:V \to V$ by

\[g(\alpha \cos x + \beta \sin x):=\frac{d}{dx}(\alpha \cos x+ \beta \sin x)=\beta \cos x -\alpha \sin x.\]
Prove that the map $g$ is a linear transformation.

**(e)** Find the matrix representation of the linear transformation $g$ with respect to the basis $\{\cos x, \sin x\}$.

(Kyoto University, Linear Algebra exam problem)

Add to solve laterLet $A$ be an $n\times n$ idempotent matrix, that is, $A^2=A$. Then prove that $A$ is diagonalizable.

Add to solve later Let $R$ be the ring of all continuous functions on the interval $[0, 2]$.

Let $I$ be the subset of $R$ defined by

\[I:=\{ f(x) \in R \mid f(1)=0\}.\]

Then prove that $I$ is an ideal of the ring $R$.

Moreover, show that $I$ is maximal and determine $R/I$.

Let $G$ be a finite group and let $N$ be a normal abelian subgroup of $G$.

Let $\Aut(N)$ be the group of automorphisms of $G$.

Suppose that the orders of groups $G/N$ and $\Aut(N)$ are relatively prime.

Then prove that $N$ is contained in the center of $G$.

Let $H$ and $K$ be normal subgroups of a group $G$.

Suppose that $H < K$ and the quotient group $G/H$ is abelian.

Then prove that $G/K$ is also an abelian group.

Let $R$ be an integral domain. Then prove that the ideal $(x^3-y^2)$ is a prime ideal in the ring $R[x, y]$.

Add to solve later**(a)** Let $f_1(x)$ and $f_2(x)$ be irreducible polynomials over a finite field $\F_p$, where $p$ is a prime number. Suppose that $f_1(x)$ and $f_2(x)$ have the same degrees. Then show that fields $\F_p[x]/(f_1(x))$ and $\F_p[x]/(f_2(x))$ are isomorphic.

**(b)** Show that the polynomials $x^3-x+1$ and $x^3-x-1$ are both irreducible polynomials over the finite field $\F_3$.

**(c)** Exhibit an explicit isomorphism between the splitting fields of $x^3-x+1$ and $x^3-x-1$ over $\F_3$.

Let $G$ be a group and let $H$ be a subgroup of finite index. Then show that there exists a normal subgroup $N$ of $G$ such that $N$ is of finite index in $G$ and $N\subset H$.

Add to solve laterLet $R$ be a commutative ring with $1$. Show that if $R$ is an integral domain, then the characteristic of $R$ is either $0$ or a prime number $p$.

Add to solve laterSuppose that $G$ is a finite group of order $p^an$, where $p$ is a prime number and $p$ does not divide $n$.

Let $N$ be a normal subgroup of $G$ such that the index $|G: N|$ is relatively prime to $p$.

Then show that $N$ contains all $p$-Sylow subgroups of $G$.

Add to solve laterLet $p$ be a prime number. Let

\[G=\{z\in \C \mid z^{p^n}=1\} \]
be the group of $p$-power roots of $1$ in $\C$.

Show that the map $\Psi:G\to G$ mapping $z$ to $z^p$ is a surjective homomorphism.

Also deduce from this that $G$ is isomorphic to a proper quotient of $G$ itself.

Let $R$ be a commutative ring with $1$. Prove that the principal ideal $(x)$ generated by the element $x$ in the polynomial ring $R[x]$ is a prime ideal if and only if $R$ is an integral domain.

Prove also that the ideal $(x)$ is a maximal ideal if and only if $R$ is a field.

Add to solve laterLet $R$ be a ring with unit $1\neq 0$.

Prove that if $M$ is an ideal of $R$ such that $R/M$ is a field, then $M$ is a maximal ideal of $R$.

(Do not assume that the ring $R$ is commutative.)

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

Add to solve laterLet $G$ 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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