Expert Help for Abstract Algebra Challenges

If a, b are elements of a ring and m, $n\in Z$, show that $(na)(mb)=(mn)(ab)$

Explain why (a) $Z_9$ is not isomorphic to $Z_3\times Z_3$;
(b) $Z_9\times Z_9$ is not isomorphic to $Z_9\times Z_3\times Z_3$.

In Exercises 1 through 4, write the set in the form $\{x|P(x)\}$, where P(x) is a property that describes the elements of the set.
1. $\{2,4,6,8,10\}$

2. $\{a,e,i,o,u\}$
3. $\{1,8,27,64,125\}$
4. $\{-2,-1,0,1,2\}$

Question 4 (Module Outcome #4): Find the best-case, worst-case and average-case number of < comparisons are performed by the following piece of pseudocode. Precondition: $n\in \{1,3,5,7,9\}\text{while}n<6\text{do}n\leftarrow n+3$

Prove the following.
(1) $Z\times 5$ is a cyclic group.
(2) $Z\times 8$ is not a cyclic group.

In the froup ${\displaystyle Z_{12}}$, find $|a|,|b|$, and $|a+b|$
$a=5,b=4$

Let H be a normal subgroup of a group G, and let $m=(G:H)$. Expose that
$a^m\in H$
for every $a\in G$

If U is a set, let ${\displaystyle G=\{X\mid X\subseteq U\}}$. Show that G is an abelian group under the operation $\oplus$ defined by ${\displaystyle X\oplus Y=\left(\frac{x}{y}\right)\cup \left(\frac{y}{x}\right)}$

Show that the prime subfield of a field of characteristic p is ringisomorphic to $Z_p$ and that the prime subfield of a field of characteristic 0 is ring-isomorphic to Q.

Let $\mathbb{R}$ sube K be a field extension of degree 2, and prove that $K\cong \mathbb{C}$. Prove that there is no field extension $\mathbb{R}$ sube K of degree 3.