There is a field with 10 elements.

There are at least three non-isomorphic rings with 4 elements.

The group $( \mathbb { Q } , + )$ is a finitely generated abelian group.

$\mathbb { Q } ( \sqrt { 7 } )$ and $\mathbb { Q } ( \sqrt { 17 } )$ are isomorphic as fields.

There is a field with 121 elements.

There are no infinite group with subgroups of index 5.

Every finite group of odd order is isomorphic to a subgroup of $A _ { n }$, the group of all even permutations.

Every group of order 6 abelian.

Two abelian groups of the same order are isomorphic.

(i) Let $G = G L \left( 2 , \mathbb { F } _ { p } \right)$. Prove that there is a Sylow $p$-subgroup $H$ of $G$ whose normalizer $N _ { G } ( H )$ is the group of all upper triangular matrices in $G$.
(ii) Hence prove that the number of Sylow subgroups of $G$ is $1 + p$.

Calculate the minimal polynomial of $\sqrt { 2 } e ^ { \frac { 2 \pi i } { 3 } }$ over $\mathbb { Q }$.

Let $G$ be a group $\mathbb { F }$ a field and $n$ a positive integer. A linear action of $G$ on $\mathbb { F } ^ { n }$ is a map $\alpha : G \times \mathbb { F } ^ { n } \rightarrow \mathbb { F } ^ { n }$ such that $\alpha ( g , v ) = \rho ( g ) v$ for some group homomorphism $\rho : G \rightarrow \mathrm { GL } _ { n } ( \mathbb { F } )$. Show that for every finite group $G$, there is an $n$ such that there is a linear action $\alpha$ of $G$ on $\mathbb { F } ^ { n }$ and such that there is a nonzero vector $v \in \mathbb { F } ^ { n }$ such that $\alpha ( g , v ) = v$ for all $g \in G$.

Let $R$ be an integral domain containing a field $F$ as a subring. Show that if $R$ is a finite-dimensional vector space over $F$, then $R$ is a field.

Pick the correct statement(s) below.
(a) There exists a group of order 44 with a subgroup isomorphic to $\mathbb { Z } / 2 \oplus \mathbb { Z } / 2$.
(b) There exists a group of order 44 with a subgroup isomorphic to $\mathbb { Z } / 4$.
(c) There exists a group of order 44 with a subgroup isomorphic to $\mathbb { Z } / 2 \oplus \mathbb { Z } / 2$ and a subgroup isomorphic to $\mathbb { Z } / 4$.
(d) There exists a group of order 44 without any subgroup isomorphic to $\mathbb { Z } / 2 \oplus \mathbb { Z } / 2$ or to $\mathbb { Z } / 4$.

Let $G$ be a group. The following statements hold.
(a) If $G$ has nontrivial centre $C$, then $G / C$ has trivial centre.
(b) If $G \neq 1$, there exists a nontrivial homomorphism $h : \mathbb { Z } \rightarrow G$.
(c) If $| G | = p ^ { 3 }$, for $p$ a prime, then $G$ is abelian.
(d) If $G$ is nonabelian, then it has a nontrivial automorphism.

Let $C [ 0,1 ]$ be the space of continuous real-valued functions on the interval $[ 0,1 ]$. This is a ring under point-wise addition and multiplication. The following are true.
(a) For any $x \in [ 0,1 ]$, the ideal $M ( x ) = \{ f \in C [ 0,1 ] \mid f ( x ) = 0 \}$ is maximal.
(b) $C [ 0,1 ]$ is an integral domain.
(c) The group of units of $C [ 0,1 ]$ is cyclic.
(d) The linear functions form a vector-space basis of $C [ 0,1 ]$ over $\mathbb { R }$.

Let $F$ be a field with 256 elements, and $f \in F [ x ]$ a polynomial with all its roots in $F$. Then,
(a) $f \neq x ^ { 15 } - 1$;
(b) $f \neq x ^ { 63 } - 1$;
(c) $f \neq x ^ { 2 } + x + 1$;
(d) if $f$ has no multiple roots, then $f$ is a factor of $x ^ { 256 } - x$.

Let $h : \mathbb { C } \rightarrow \mathbb { C }$ be an analytic function such that $h ( 0 ) = 0 ; h \left( \frac { 1 } { 2 } \right) = 5$, and $| h ( z ) | < 10$ for $| z | < 1$. Then,
(a) the set $\{ z : | h ( z ) | = 5 \}$ is unbounded by the Maximum Principle;
(b) the set $\left\{ z : \left| h ^ { \prime } ( z ) \right| = 5 \right\}$ is a circle of strictly positive radius;
(c) $h ( 1 ) = 10$;
(d) regardless of what $h ^ { \prime }$ is, $h ^ { \prime \prime } \equiv 0$.

Let $$G = \left\{\left(\begin{array}{cc}a & b \\ 0 & a^{-1}\end{array}\right) : a, b \in \mathbb{R}, a > 0\right\}, \quad N = \left\{\left(\begin{array}{cc}1 & b \\ 0 & 1\end{array}\right) : b \in \mathbb{R}\right\}.$$ Which of the following are true?
(A) $G/N$ is isomorphic to $\mathbb{R}$ under addition.
(B) $G/N$ is isomorphic to $\{a \in \mathbb{R} : a > 0\}$ under multiplication.
(C) There is a proper normal subgroup $N'$ of $G$ which properly contains $N$.
(D) $N$ is isomorphic to $\mathbb{R}$ under addition.

Let $m$ and $n$ be positive integers and $p$ a prime number. Let $G \subseteq \mathrm{GL}_{m}(\mathbb{F}_{p})$ be a subgroup of order $p^{n}$. Let $U \subseteq \mathrm{GL}_{m}(\mathbb{F}_{p})$ be the subgroup that consists of all the matrices with 1's on the diagonal and 0's below the diagonal. Show that there exists $A \in \mathrm{GL}_{m}(\mathbb{F}_{p})$ such that $AGA^{-1} \subseteq U$.

What is the cardinality of the centre of $O_2(\mathbb{R})$? (Definition: The centre of a group $G$ is $\{g \in G \mid gh = hg \text{ for every } h \in G\}$. Hint: Reflection matrices and permutation matrices are orthogonal.)
(A) 1;
(B) 2;
(C) The cardinality of $\mathbb{N}$;
(D) The cardinality of $\mathbb{R}$.

A subgroup $H$ of a group $G$ is said to be a characteristic subgroup if $\sigma(H) = H$ for every group isomorphism $\sigma : G \longrightarrow G$ of $G$.
(A) Determine all the characteristic subgroups of $(\mathbb{Q}, +)$ (the additive group).
(B) Show that every characteristic subgroup of $G$ is normal in $G$. Determine whether the converse is true.

Write $V$ for the space of $3 \times 3$ skew-symmetric real matrices.
(A) Show that for $A \in SO_3(\mathbb{R})$ and $M \in V$, $AMA^t \in V$. Write $A \cdot M$ for this action.
(B) Let $\Phi : \mathbb{R}^3 \longrightarrow V$ be the map $$\begin{bmatrix} u \\ v \\ w \end{bmatrix} \mapsto \begin{bmatrix} 0 & w & -v \\ -w & 0 & u \\ v & -u & 0 \end{bmatrix}$$ With the usual action of $SO_3(\mathbb{R})$ on $\mathbb{R}^3$ and the above action on $V$, show that $\Phi(Av) = A \cdot \Phi(v)$ for every $A \in SO_3(\mathbb{R})$ and $v \in \mathbb{R}^3$.
(C) Show that there does not exist $M \in V$, $M \neq 0$ such that for every $A \in SO_3(\mathbb{R})$, $A \cdot M$ belongs to the span of $M$.

Let $\mathbb{k}$ be a field, $n$ a positive integer and $G$ a finite subgroup of $\mathrm{GL}_n(\mathbb{k})$ such that $|G| > 1$. Further assume that every $g \in G$ is upper-triangular and all the diagonal entries of $g$ are 1.
(A) Show that $\operatorname{char}\,\mathbb{k} > 0$. (Hint: consider the minimal polynomials of elements of $G$.)
(B) Show that the order of $g$ is a power of $\operatorname{char}\,\mathbb{k}$, for every $g \in G$.
(C) Show that the centre of $G$ has at least two elements.

For a field $F$, $F^{\times}$ denotes the multiplicative group ($F \backslash \{0\}, \times$). Choose the correct statement(s) from below:
(A) Every finite subgroup of $\mathbb{R}^{\times}$ is cyclic;
(B) The order of every non-trivial finite subgroup of $\mathbb{R}^{\times}$ is a prime number;
(C) There are infinitely many non-isomorphic non-trivial finite subgroups of $\mathbb{R}^{\times}$;
(D) The order of every non-trivial finite subgroup of $\mathbb{C}^{\times}$ is a prime number.