Hey I'm having some trouble with this question

Heres what I've come up with so far.Deduce that any group of order 2p. where p is a prime p>2 Must contain a subgroup of order p

Since G is closed this implies that for every element

where q is a positive integer. Hence each element of g generates a cyclic subgroup of G of order q.

By Lagranges therom q can only be 2 or p.

The trouble I'm having now is proving G cannot just consist of elements of order 2 (or did I misunderstand the question?)