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

42

SUBSTITUTION GROUPS

[CH. II

one of these is of order 4 while the other is of order 2. Their
substitutions are as follows:
1, ab, cd, ab cd; 1, ab.cd.
According to Sylow's theorem, (abcd)all involves at least
one subgroup of order 8 and all its subgroups of this order are
conjugate. Since all conjugate groups are regarded as identical
in the enumeration of groups, there is one and only one substitution group of degree 4 and of order 8. This is known
to be transitive (~ 2). We shall represent it by the symbol
(abcd)s. Since the order of every transitive group is a multiple of its degree, and since a group must be included in the
symmetric group of its own degree, it results that the order
of a transitive group of degree four is 4, 8, 12 or 24.
As there is one and only one such group of each of the orders
8 and 24, it remains to determine all the possible groups of
orders 4 and 12. We know that there are two transitive groups
of the former order; viz., the subgroups of the octic group considered in ~ 2, and there is one group of the latter order; viz.,
(abcd)pos.* We proceed to prove that no other groups of these
orders are possible. Another transitive group of order 4 would
also be regular, since the average number of letters in its substitutions is 3. It could not be cyclic, since there is one and only
one cyclic group of each order, and two simply isomorphic
regular groups are conjugate. If it were non-cyclic it would
involve all the possible substitutions of the form ab cd in the
symmetric group of degree 4. This proves that there are only
two regular groups of degree 4. One of these has three conjugates under the symmetric group while the other is invariant
under this group.
To prove that there is only one group of order 12 and degree
4, we observe that every such group would have to contain
a subgroup of order 3 according to Sylow's theorem. As all
the subgroups of order 3 in (abcd)all are conjugate, we may
assume that every group of order 12 and degree 4 includes
(abc). Since this subgroup could not be invariant under a tran* The symbol (ala2... an)pos represents the alternating group of degree n.