Theory and applications of finite groups, by G.A. Miller, H. F. Blichfeldt [and] L. E. Dickson.

62

ABSTRACT GROUPS

[CH. III

If n is odd, these substitutions can be selected as follows:
S1 =ala2' a3a4.. n-2an-1
2= a2a3.. 'a,-3a,-2' a-la
Since the product of s1S2 in each case is a cyclic substitution
of order n it results that si and s2 generate the dihedral group
of order 2n.
The non-cyclic group of order 4 is the only dihedral group
which does not involve non-commutative elements. A group
which contains no non-commutative elements is called commutative or abelian. Since there is one cyclic group of every
order and one dihedral group of every even order greater than
2, there must be at least two groups of every even order greater
than 2. When this order exceeds 4 one of these two groups,
whose existence has been proved here, is abelian while the other
is non-abelian.
Instead of defining the dihedral group of order 2n as the
group generated by two elements of order 2 whose product is
of order n, it could also have been defined as the group generated by a cyclic group H of order n and an element of order
2 which transforms every element of H into its inverse. Both
of these definitions of the dihedral group are very useful. If
n is even we can find an element t of order 4 which transforms
every element of H into its inverse and has its square in H.
The group of order 2n generated by H and this t is called the
dicyclic group whenever n> 2, and there is one and only one
such group of every order which is divisible by 4 and exceeds
4. The smallest dicyclic group is the group of order 8 generated
by the four quaternion units 1, i, j, k. This is known as the
quaternion group. Its properties were studied by W. R. Hamilton.
The fact that there is no more than one dicyclic group of a
given order can be at once proved by proving that two such
groups of the same order are simply isomorphic. The existence of this group for every even value of n may be proved
by means of substitution groups as follows: Write a transitive
dihedral group of order 2n on two distinct sets of letters and