Sample final

These mock exams are provided to help you prepare for Term/Final tests. The best way to use these practice tests is to try the problems as if you were taking the test. Please don't look at the solution until you have attempted the question(s). Only reading through the answers or studying them, will typically not be helpful in preparing since it is too easy to convince yourself that you understand them.  

Exercise $\PageIndex{1}$

 

Consider the symmetric group $S_7$.

1. Factor the permutation $(1 2 3 4 5)(67)(1357)(613)$ into disjoint cycles.
2. Identify all possible types of permutations and specify the quantity of each type.
3.  Find the possible order of all the elements of $S_7$.
4. Show that $S_7$ is not abelian.

Exercise $\PageIndex{2}$

1. Show that $A_4$ is normal to $S_4$.
2.  What are the conjugacy classes of $A_4$?

Exercise $\PageIndex{3}$

Consider $D_4=\{e, a, a^2,a^3, b, ab, a^2b, a^3b\}$ with $a^4=e, b^2=e, ba=a^3b.$

1. Show that $Z(D_4)=\{e, a^2\}.$ (Hint: $Z(G)=\{x \in G| ax=xa, \forall a\in G\}.$)
2. List the left and right cosets of $Z(D_4),$ in $D_4.$  

Exercise $\PageIndex{4}$

1. Prove or disprove: $U(5)$ is isormorphic to $\mathbb{Z}_4$ .
2.  Prove or disporove: $\mathbb{Z}$ under addition is isormorphic to $\mathbb{Q}$ under addition.

Exercise $\PageIndex{5}$

1. Show that $17$ and $60$ are relatively prime, determine the multiplicative inverse of $[17]$ in $\mathbb{Z}_{60}.$
2.  Find all $x \in \mathbb{Z}$ satisfying the equation $17x \equiv 12(mod \,60).$

Exercise $\PageIndex{6}$

Let $G$ be a group. Prove or disprove the following statements:

Let $H$ and $K$ be subgroups of the group $G$. Then

1.  $H\cup K$ is a subgroup of $G$.
2.  $H\cap K$ is a subgroup of $G$.

Exercise $\PageIndex{7}$

Consider the group $\mathbb{Z}_{12}$ with addition modulo $12.$  

1.  Is $(\mathbb{Z}_{12} ,+ mod (12) )$ cyclic group?  If so, what are the possible generators? How many distinct generators exist? What can you say about the number of generators, considering the Euler totient function and the possible generators in relation to divisors?
2.  Let $G=\langle g \rangle$ be a cyclic group with $|g|=n$. Then  $G=\langle g^k \rangle$ if and only if $gcd(k,n)=1.$
3.  Find all subgroups of $\mathbb{Z}_{12}$ generated by each element. What are their respective orders? What can you say about these orders?

Exercise $\PageIndex{8}$

1.  Let $G=<a>$ where   $|a|=12,$ and let $H=<a^4>.$ Find all cosets in $G/H$ and  write down the Cayley table. Is $G/H$ cyclic? Why or why not?
     Find the cosets in $D_6/Z(D_6),$ write down the Cayley table of  $D_6/Z(D_6).$  Is $D_6/Z(D_6)$ cyclic, why or why not?

2.  Prove or disprove the following statements:  If $H$ is a normal subgroup of $G$ such that $H$ and $G/H$ are abelian, then 
   $G$ is abelian.

Exercise $\PageIndex{9}$

An element $e$ of a ring $R$ is said to be idempotent if $e^2=e.$

1.   If $e$ is an idempotent of a ring $R$, then show that $1-2e$ is a unit.
2.  Find all idempotents in $\mathbb{Z}_{12}$.
3.   Find all the idempotents in an integral domain $R$.

Exercise $\PageIndex{10}$

 List or characterize all of the following rings' units. Justify your answer.

1.   $\mathbb{Z}_{12}$
2. $M_{22}(\mathbb{Z}_2)$
3. $\mathbb{Z}(i)$