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.