Is the Set of Nilpotent Element an Ideal?
Problem 620
Is it true that a set of nilpotent elements in a ring $R$ is an ideal of $R$?
If so, prove it. Otherwise give a counterexample.
Add to solve later Tagged: abelian group
Abelian Group Problems and Solutions.
The other popular topics in Group Theory are:
Add to solve later The Set of Square Elements in the Multiplicative Group $(\Zmod{p})^*$
Problem 616
Suppose that $p$ is a prime number greater than $3$.
Consider the multiplicative group $G=(\Zmod{p})^*$ of order $p-1$.
(a) Prove that the set of squares $S=\{x^2\mid x\in G\}$ is a subgroup of the multiplicative group $G$.
(b) Determine the index $[G : S]$.
(c) Assume that $-1\notin S$. Then prove that for each $a\in G$ we have either $a\in S$ or $-a\in S$.
Add to solve later If a Half of a Group are Elements of Order 2, then the Rest form an Abelian Normal Subgroup of Odd Order
Problem 575
Let $G$ be a finite group of order $2n$.
Suppose that exactly a half of $G$ consists of elements of order $2$ and the rest forms a subgroup.
Namely, suppose that $G=S\sqcup H$, where $S$ is the set of all elements of order in $G$, and $H$ is a subgroup of $G$. The cardinalities of $S$ and $H$ are both $n$.
Then prove that $H$ is an abelian normal subgroup of odd order.
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 Elements of Finite Order of an Abelian Group form a Subgroup
Problem 522
Let $G$ be an abelian group and let $H$ be the subset of $G$ consisting of all elements of $G$ of finite order. That is,
\[H=\{ a\in G \mid \text{the order of $a$ is finite}\}.\]
Prove that $H$ is a subgroup of $G$.
Add to solve later The Existence of an Element in an Abelian Group of Order the Least Common Multiple of Two Elements
Problem 497
Let $G$ be an abelian group.
Let $a$ and $b$ be elements in $G$ of order $m$ and $n$, respectively.
Prove that there exists an element $c$ in $G$ such that the order of $c$ is the least common multiple of $m$ and $n$.
Also determine whether the statement is true if $G$ is a non-abelian group.
Add to solve later Every Finite Group Having More than Two Elements Has a Nontrivial Automorphism
Problem 495
Prove that every finite group having more than two elements has a nontrivial automorphism.
(Michigan State University, Abstract Algebra Qualifying Exam)
Read solution
Add to solve later Every Group of Order 20449 is an Abelian Group
A Module is Irreducible if and only if It is a Cyclic Module With Any Nonzero Element as Generator
Problem 434
Let $R$ be a ring with $1$.
A nonzero $R$-module $M$ is called irreducible if $0$ and $M$ are the only submodules of $M$.
(It is also called a simple module.)
(a) Prove that a nonzero $R$-module $M$ is irreducible if and only if $M$ is a cyclic module with any nonzero element as its generator.
(b) Determine all the irreducible $\Z$-modules.
Add to solve later Fundamental Theorem of Finitely Generated Abelian Groups and its application
Problem 420
In this post, we study the Fundamental Theorem of Finitely Generated Abelian Groups, and as an application we solve the following problem.
Problem.
Let $G$ be a finite abelian group of order $n$.
If $n$ is the product of distinct prime numbers, then prove that $G$ is isomorphic to the cyclic group $Z_n=\Zmod{n}$ of order $n$.
Add to solve later Can $\Z$-Module Structure of Abelian Group Extend to $\Q$-Module Structure?
Problem 418
If $M$ is a finite abelian group, then $M$ is naturally a $\Z$-module.
Can this action be extended to make $M$ into a $\Q$-module?
Add to solve later The Inverse Image of an Ideal by a Ring Homomorphism is an Ideal
Problem 411
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 $R$.
Add to solve later Prove a Group is Abelian if $(ab)^3=a^3b^3$ and No Elements of Order $3$
Problem 402
Let $G$ be a group. Suppose that we have
\[(ab)^3=a^3b^3\]
for any elements $a, b$ in $G$. Also suppose that $G$ has no elements of order $3$.
Then prove that $G$ is an abelian group.
Add to solve later Prove a Group is Abelian if $(ab)^2=a^2b^2$
Problem 401
Let $G$ be a group. Suppose that
\[(ab)^2=a^2b^2\]
for any elements $a, b$ in $G$. Prove that $G$ is an abelian group.
Add to solve later Order of Product of Two Elements in a Group
Problem 354
Let $G$ be a group. Let $a$ and $b$ be elements of $G$.
If the order of $a, b$ are $m, n$ respectively, then is it true that the order of the product $ab$ divides $mn$? If so give a proof. If not, give a counterexample.
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 Surjective Group Homomorphism to $\Z$ and Direct Product of Abelian Groups
Problem 342
Let $G$ be an abelian group and let $f: G\to \Z$ be a surjective group homomorphism.
Prove that we have an isomorphism of groups:
\[G \cong \ker(f)\times \Z.\]
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 Quotient Group of Abelian Group is Abelian
Problem 340
Let $G$ be an abelian group and let $N$ be a normal subgroup of $G$.
Then prove that the quotient group $G/N$ is also an abelian group.
Add to solve later 
