You solved 0 problems!!
Prove that every group of order $24$ has a normal subgroup of order $4$ or $8$.
Let $F$ be a finite field.
Prove that each element in the field $F$ is the sum of two squares in $F$.
Prove that any field automorphism of the field of real numbers $\R$ must be the identity automorphism.
Let $G$ and $H$ be groups and let $\phi: G \to H$ be a group homomorphism.
Suppose that $f:G\to H$ is bijective.
\[\psi \circ \phi=\id_G \text{ and } \phi \circ \psi=\id_H.\] Then prove that $\psi:H \to G$ is also a group homomorphism.
Let $G, G’$ be groups. Let $\phi:G\to G’$ be a group homomorphism.
Let $A=B=\Z$ be the additive group of integers.
Define a map $\phi: A\to B$ by sending $n$ to $2n$ for any integer $n\in A$.
(a) Prove that $\phi$ is a group homomorphism.
(b) Prove that $\phi$ is injective.
(c) Prove that there does not exist a group homomorphism $\psi:B \to A$ such that $\psi \circ \phi=\id_A$.
(a) Let $R$ be a commutative ring. If we regard $R$ as a left $R$-module, then prove that any two distinct elements of the module $R$ are linearly dependent.
(b) Let $f: M\to M’$ be a left $R$-module homomorphism. Let $\{x_1, \dots, x_n\}$ be a subset in $M$. Prove that if the set $\{f(x_1), \dots, f(x_n)\}$ is linearly independent, then the set $\{x_1, \dots, x_n\}$ is also linearly independent.
Suppose that $f:R\to R’$ is a surjective ring homomorphism.
Prove that if $R$ is a Noetherian ring, then so is $R’$.
Let $f: R\to R’$ be a ring homomorphism. Let $P$ be a prime ideal of the ring $R’$.
Prove that the preimage $f^{-1}(P)$ is a prime ideal of $R$.
Let $f:R\to R’$ be a ring homomorphism. Let $I’$ be an ideal of $R’$ and let $I=f^{-1}(I)$ be the preimage of $I$ by $f$. Prove that $I$ is an ideal of the ring $R$.
Let $G$ be a finite group of order $21$ and let $K$ be a finite group of order $49$.
Suppose that $G$ does not have a normal subgroup of order $3$.
Then determine all group homomorphisms from $G$ to $K$.
Let $R$ be the ring of all continuous functions on the interval $[0, 2]$.
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 $G$ be an abelian group and let $f: G\to \Z$ be a surjective group homomorphism.
Let $G$ be a group and define a map $f:G\to G$ by $f(a)=a^2$ for each $a\in G$.
Then prove that $G$ is an abelian group if and only if the map $f$ is a group homomorphism.
Let $R$ be a ring with unity.
Suppose that $f$ and $g$ are ring homomorphisms from $\Q$ to $R$ such that $f(n)=g(n)$ for any integer $n$.
Then prove that $f=g$.
\[\epsilon(\sum_{i=1}^na_i g_i)=\sum_{i=1}^na_i,\] where $a_i\in R$ and $G=\{g_i\}_{i=1}^n$, is a ring homomorphism, called the augmentation map and the kernel of $\epsilon$ is called the augmentation ideal.
(a) Prove that the augmentation ideal in the group ring $RG$ is generated by $\{g-e \mid g\in G\}$.
(b) Prove that if $G=\langle g\rangle$ is a finite cyclic group generated by $g$, then the augmentation ideal is generated by $g-e$.
\end{bmatrix} \quad \middle| \quad \text{ for any} a,b,c\in F\, \right\}\] be the Heisenberg group over $F$.
Determine which matrices lie in the center of $H(F)$ and prove that the center $Z\big(H(F)\big)$ is isomorphic to the additive group $F$.
\[\phi(e, g)=g \text{ and } \phi(g, e)=g, \tag{*}\] for any $g\in G$.
Then prove that $\phi=\mu$.
Also prove that the group $G$ is abelian.
Let $\Z$ be the ring of integers and let $R$ be a ring with unity.
Determine all the ring homomorphisms from $\Z$ to $R$.
I’m Yu Tsumura.
The list of linear algebra problems is available here.
Enter your email address to subscribe to this blog and receive notifications of new posts by email.
Mathematical equations are created by MathJax. See How to use MathJax in WordPress if you want to write a mathematical blog.
Find All Values of $a$ which Will Guarantee that $A$ Has Eigenvalues 0, 3, and -3.
Is the Set of Nilpotent Element an Ideal?
Number Theoretical Problem Proved by Group Theory. $a^{2^n}+b^{2^n}\equiv 0 \pmod{p}$ Implies $2^{n+1}|p-1$.
How to Diagonalize a Matrix. Step by Step Explanation.
Enter your email address to subscribe to this blog and receive notifications of new posts by email.
If you are a member, Login here.
Problems in Mathematics © 2019. All Rights Reserved.
