Groups of order 2p. In this series of exercises we will classify all groups of order 2p, where p is an odd prime.
((a)) Assume G is a group of order 2p, where p is an odd prime. If a∈G, show that a must have order 1, 2, p, or 2p.
((b)) Suppose that G has an element of order 2p. Prove that G is isomorphic to Z2p. Hence, G is cyclic.
((c)) Suppose that G does not contain an element of order 2p. Show that G must contain an element of order p. Hint: Assume that G does not contain an element of order p.
((d)) Suppose that G does not contain an element of order 2p. Show that G must contain an element of order 2.
((e)) Let P be a subgroup of G with order p and y∈G have order 2. Show that yP=Py.
((f)) Suppose that G does not contain an element of order 2p and P=⟨z⟩ is a subgroup of order p generated by z. If y is an element of order 2, then yz=zky for some 2≤k<p.
((g)) Suppose that G does not contain an element of order 2p. Prove that G is not abelian.
((h)) Suppose that G does not contain an element of order 2p and P=⟨z⟩ is a subgroup of order p generated by z and y is an element of order 2. Show that we can list the elements of G as {ziyj∣0≤i<p,0≤j<2}.
((i)) Suppose that G does not contain an element of order 2p and P=⟨z⟩ is a subgroup of order p generated by z and y is an element of order 2. Prove that the product (ziyj)(zrys) can be expressed uniquely as zmyn for some non-negative integers m,n. Thus, conclude that there is only one possibility for a non-abelian group of order 2p, it must therefore be the one we have seen already, the dihedral group.