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# Groups of order 2p

Let be a group, a prime number and . Prove that either is cyclic or where is the dihedral group of order .

Solution

It is clear for . So we will assume that is an odd prime. Choose with . Let . Since every subgroup of index is normal, is a normal subgroup of and thus

for some integer . Note that, since is odd, and hence

Now

because . Thus and hence because . Therefore either or . So we will consider two cases:

• . In this case

and so . Thus is abelian and hence because is odd. So is cyclic in this case.

•  . In this case

and so