## 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 laterof the day

Abelian Group Problems and Solutions.

The other popular topics in Group Theory are:

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 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 $-a\in S$.

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 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 $G$ 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 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 laterProve that every finite group having more than two elements has a nontrivial automorphism.

(*Michigan State University, Abstract Algebra Qualifying Exam*)

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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.

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 $n$.

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 laterLet $f:R\to R’$ 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 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 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.

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.

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.

Prove that we have an isomorphism of groups:

\[G \cong \ker(f)\times \Z.\]

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 $G$ 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.