Let $m$ be a positive integer, which is not a perfect square. Show that the mapping $\alpha + \beta \sqrt{m} \mapsto \alpha - \beta \sqrt{m}, \alpha ,\beta \in \mathbb{Q}$ , is an automorphism of $\mathbb{Q} (\sqrt{m})$ . Hence show that for any rational numbers $\alpha , \beta$ with $\beta \neq 0$ , the minimal polynomials of $\alpha + \beta \sqrt{m}$ and $\alpha - \beta \sqrt{m}$ over $\mathbb{Q}$ are same .

Let $F$ be a field and $\alpha$ is algebraic over $F$. Show that if $[F(\alpha):F)]$ is odd, then $F(\alpha) =F(\alpha ^2)$ .

Give an example of a semigroup $S$ which has a left identity and in which every element has a right inverse but  $S$ is not a group.

Show that if $G$ is a group of order $2n$ , then there are exactly an odd number of elements of order $2$ in $G$ . Also show that if n is odd and $G$ is abelian, there is only one element of order $2$ .

Let $G$ be the group of all $2 \times 2$ matrices $\left( \begin{array}{cols} a&b \\ c&d \\ \end{array} \right)$, where $a,b,c,d$ are integers modulo $p,p$ a prime number, such that $ad - bc \neq 0$. Determine the order of $G$ .

Show that, in a group $G, \circ(ab) = \circ(ba)$ for all $a,b \in G$ . Further show that if $a$ and $b$ commute, and $\circ(a) =n_1 , \circ(b) =n_2$ and $(n_1,n_2) =1$, then $\circ(ab)= \circ(ba)=n_1 n_2$ .

Let $G$ be a group, $H \le G$ and $g \in G$ . Show that if $\circ(g) =n$ and $g^m \in H$, where $n$ and $m$ are coprime integers, then $g \in H$ .

Let $G$ be a group, $H \le G$ and $g \in G$. Show that if $\circ(g) =n_1n_2$ and $g^m \in H$, where $n$ and $m$ are coprime integers, then $g \in H$ .

Let $G$ be a group and $g \in G$ with $\circ(g) =n_1n_2$ , where $n_1$ and $n_2$ are coprime positive integers. Then show that there are elements $g_1,g_2 \in G$ such that $g = g_1g_2 =g_2g_1$ and $\circ(g_1) =n_1 , \circ(g_2) =n_2$ . Further show that $g_1$ and $g_2$ are uniquely determined by these conditions.

Let $G$ be a group such that the intersection of all its subgroups which are different from $\{e\}$ is a subgroup different from $\{e\}$ . Prove that every element in $G$ has finite order.

If a group $G \neq \{e\}$ has no nontrivial subgroups, show that $G$ must be a finite group of prime order.