Let $G$ be a group of order $57$. Assume that $G$ is not a cyclic group.
Then determine the number of elements in $G$ of order $3$.
Read solution
Add to solve later Let $G$ be a group. Suppose that the number of elements in $G$ of order $5$ is $28$.
Determine the number of distinct subgroups of $G$ of order $5$.
Add to solve later Let $G$ be a finite group. Let $S$ be the set of elements $g$ such that $g^5=e$, where $e$ is the identity element in the group $G$.
Prove that the number of elements in $S$ is odd.
Add to solve later Let $m$ and $n$ be positive integers such that $m \mid n$.
(a) Prove that the map $\phi:\Zmod{n} \to \Zmod{m}$ sending $a+n\Z$ to $a+m\Z$ for any $a\in \Z$ is well-defined.
(b) Prove that $\phi$ is a group homomorphism.
(c) Prove that $\phi$ is surjective.
(d) Determine the group structure of the kernel of $\phi$.
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 $\F_3=\Zmod{3}$ be the finite field of order $3$.
Consider the ring $\F_3[x]$ of polynomial over $\F_3$ and its ideal $I=(x^2+1)$ generated by $x^2+1\in \F_3[x]$.
(a) Prove that the quotient ring $\F_3[x]/(x^2+1)$ is a field. How many elements does the field have?
(b) Let $ax+b+I$ be a nonzero element of the field $\F_3[x]/(x^2+1)$, where $a, b \in \F_3$. Find the inverse of $ax+b+I$.
(c) Recall that the multiplicative group of nonzero elements of a field is a cyclic group.
Confirm that the element $x$ is not a generator of $E^{\times}$, where $E=\F_3[x]/(x^2+1)$ but $x+1$ is a generator.
Add to solve later Let $R$ be a ring with unit $1$. Suppose that the order of $R$ is $|R|=p^2$ for some prime number $p$.
Then prove that $R$ is a commutative ring.
Add to solve later Let $\Q=(\Q, +)$ be the additive group of rational numbers.
(a) Prove that every finitely generated subgroup of $(\Q, +)$ is cyclic.
(b) Prove that $\Q$ and $\Q \times \Q$ are not isomorphic as groups.
Add to solve later Let $G$ be a finite group of order $217$.
(a) Prove that $G$ is a cyclic group.
(b) Determine the number of generators of the group $G$.
Add to solve later 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 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 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 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 Let $R$ be a commutative ring with $1$ and let $G$ be a finite group with identity element $e$. Let $RG$ be the group ring. Then the map $\epsilon: RG \to R$ defined by
\[\epsilon(\sum_{i=1}^na_i g_i)=\sum_{i=1}^na_i,\]
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 $g-e$.
Read solution
Add to solve later Let $G$ be a group. (Do not assume that $G$ is a finite group.)
Prove that $G$ is a simple abelian group if and only if the order of $G$ is a prime number.
Add to solve later Show that $\Q(\sqrt{2+\sqrt{2}})$ is a cyclic quartic field, that is, it is a Galois extension of degree $4$ with cyclic Galois group.
Add to solve later Show that a group $G$ is cyclic if and only if there exists a surjective group homomorphism from the additive group $\Z$ of integers to the group $G$.
Add to solve later Let $G$ be an infinite cyclic group. Then show that $G$ does not have a composition series.
Add to solve later 
