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.

Give an example of a group \(G\) having a subgroup \(H\) and two elements \(a,b\) such that \(Ha=Hb\) but \(aH \neq bH\) .

Let \(H \le G\), and \(a \in G\) . Let \(aHa^{-1} = \{aha^{-1} |h \in H \}\) . Show that \(aHa^{-1}\) is a subgroup of \(G\).

Suppose that \(H \le G\) such that whenever \(Ha \neq Hb\) then \(aH \neq bH\) . Prove that \(gHg^{-1} \subseteq H\) for all \(g \in G\) .

If \(H\) and \(K\) are subgroups of orders \(p\) and \(n\) , respectively, where \(p\) is a prime number, show that either \(H \cap K=\{e\}\) or \(H \le K\) .