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 later Tagged: isomorphism theorem
Add to solve later If the Quotient is an Infinite Cyclic Group, then Exists a Normal Subgroup of Index $n$
Problem 557
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 The Quotient Ring by an Ideal of a Ring of Some Matrices is Isomorphic to $\Q$.
Problem 525
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 $\Q$.
Add to solve later Prove the Ring Isomorphism $R[x,y]/(x) \cong R[y]$
Problem 517
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 A Module $M$ is Irreducible if and only if $M$ is isomorphic to $R/I$ for a Maximal Ideal $I$.
Problem 449
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.
Add to solve later Subspace Spanned By Cosine and Sine Functions
Problem 435
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 later Idempotent Matrices are Diagonalizable
Problem 429
Let $A$ be an $n\times n$ idempotent matrix, that is, $A^2=A$. Then prove that $A$ is diagonalizable.
Add to solve later A Maximal Ideal in the Ring of Continuous Functions and a Quotient Ring
Problem 345
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 $R/I$.
Add to solve later Abelian Normal subgroup, Quotient Group, and Automorphism Group
Problem 343
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 $G$.
Add to solve later If Quotient $G/H$ is Abelian Group and $H < K \triangleleft G$, then $G/K$ is Abelian
Problem 341
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.
Add to solve later $(x^3-y^2)$ is a Prime Ideal in the Ring $R[x, y]$, $R$ is an Integral Domain.
Problem 239
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 Explicit Field Isomorphism of Finite Fields
Problem 233
(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 $\F_3$.
Add to solve later Subgroup of Finite Index Contains a Normal Subgroup of Finite Index
Problem 232
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 later Characteristic of an Integral Domain is 0 or a Prime Number
Problem 228
Let $R$ 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 later Subgroup Containing All $p$-Sylow Subgroups of a Group
Problem 227
Suppose 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 later Group of $p$-Power Roots of 1 is Isomorphic to a Proper Quotient of Itself
Problem 221
Let $p$ 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.
Add to solve later The Ideal $(x)$ is Prime in the Polynomial Ring $R[x]$ if and only if the Ring $R$ is an Integral Domain
Problem 198
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 later If the Quotient Ring is a Field, then the Ideal is Maximal
Problem 197
Let $R$ 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.)
Add to solve later 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 later A Subgroup of the Smallest Prime Divisor Index of a Group is Normal
Problem 105
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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