Abstract Algebra

Determine if the function $f(x) = 5x + 2$ is surjective from the integers to the integers.

Find the inverse of the function $f(x) = 5x + 2$.

Given a group $G$ whose order is 36, demonstrate why $G$ is not simple by using the extended Cayley theorem and considerations of cosets and homomorphisms.

Let $G$ be a finite group. Prove that $G$ is isomorphic to a group of permutations.

Prove the two properties: 1) If $aH = bH$, then $a \in bH$. 2) If $a \in bH$, then $aH = bH$.

Are all cyclic groups Abelian? And are all Abelian groups cyclic?

Find a counterexample of a cyclic group that is not Abelian or an Abelian group that is not cyclic.

Every cyclic group is Abelian. Prove it.

Prove that every cyclic group is abelian by taking two arbitrary elements from the group and showing that they commute.

Determine the order of the direct sum $\mathbb{Z}/4\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}$ and determine if this group is cyclic.

Find integers $s$ and $t$ that are in $\\\\Z$ (the set of integers) such that the linear combination $1224s + 567t = 9$.

Let G be the group of integers under addition and H be the subgroup 3ℤ. What are the cosets of H in G? Construct the Cayley table for the factor group G/H.

Show that the group $\mathbb{Z}_7 \times$ is a cyclic group. Find all its generators, proper subgroups, and the order of every element.

Given a finite group G and a simple group S, find all groups G where N is a normal subgroup of G and their quotient is a simple group S.

Find examples of commutative rings with one that are not integral domains.

Find examples of integral domains that are not unique factorization domains (UFDs).

Find examples of UFDs that are not PIDs (Principal Ideal Domains).

Find a PID that is not a Euclidean domain.

Identify a Euclidean domain that is not a field.

Explain the role of irreducible and prime elements in the context of commutative rings with one.