Answer Happy

You will produce an account of the classification of the
isomorphism classes of all groups of order 2p, where p is an odd
prime (i.e. p 6= 2). The parts below describe a series of
intermediate results that when put together achieves the
classification of such groups. You should work to prove each one in
turn and present the final complete argument in your submission,
using references to results from AATA as appropriate.
Partial credit can be given where you have proved some of the
intermediate results, but not others. If you cannot prove a
particular part, you can assume it when trying to prove later
parts.
Assume G is a group of order 2p, where p is an odd prime.
(a) 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 and derive a contradiction by
showing that G must be abelian and investigating its subgroups.
(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 = P y.
(f) Suppose that G does not contain an element of order 2p and P
= hzi is a subgroup of order p generated by z. If y is an element
of order 2, then yz = z ky for some 2 ≤ k < p. Hint: Rule out
the k = 0, 1 cases by considering the element zy and its order
(g) Suppose that G does not contain an element of order 2p and P
= hzi 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 {z iy j | 0
≤ i < p, 0 ≤ j < 2}
(h) Suppose that G does not contain an element of order 2p and P
= hzi is a subgroup of order p generated by z and y is an element
of order 2. Now show that in the relation yz = z ky it must in fact
be that k = p − 1. Hence argue that such a G is in fact isomorphic
to Dp, the dihedral group of order 2p, as described in the
introduction to question 1. Hint: Do this by carefully considering
the possible orders of the product element yz.