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

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GROUPS ON FOUR LETTERS

41

EXERCISES
1. If all the substitutions of order m, m>2, in a group are conjugate,
they must be transformed under the group according to an imprimitive
group.
Suggestion: The generating substitutions of one cyclic subgroup are
transformed into all of those of another.
2. If an imprimitive group contains substitutions besides the identity
which do not interchange any of its systems of imprimitivity, in a given
set of systems of imprimitivity, all such substitutions constitute an invariant subgroup.
3. Every regular group of composite order is imprimitive, and involves
as many different sets of systems as it has subgroups, excluding the identity.
4. A transitive group of order p", p being a prime number and a> 1,
is always imprimitive.
Suggestion: Consider its complete sets of congjuate substitutions
and observe that each of these sets involves pa distinct substitutions.
5. Every invariant subgroup besides the identity of a primitive group
is transitive.
6. The total number of substitutions which are commutative with
every substitution of the intransitive group obtained by establishing a
simple isomorphism between n, n>2, symmetric groups of degree n,
written in distinct sets of letters, constitute a conjugate intransitive group.
16. Groups Involving no More than Four Letters. The
only possible substitution group on two letters is 1, ab. Since
every system of intransitivity must involve at least two letters,
a group of degree 3 is necessarily transitive. It is also included
in (abc)all,* and hence its order is a divisor of 6. Therefore
(abc)all and (abc) are the only two possible groups of degree 3.
If a group of degree 4 is intransitive, each of its two systems of
intransitivity must be of degree 2. The largest intransitive
group of degree 4 is therefore the direct product of (ab), (cd).
The other possible intransitive group is a simple isomorphism
between the substitutions of these transitive groups of degree 2.
Hence there are two and only two intransitive groups of degree 4;
* The notation (ala2... an)all is used to represent the symmetric group of
degree n, while (aa2... an) represents the group generated by the cyclic substitution aa2... an. It should, however, be emphasized that the symbol
(ala2... an) is also often employed to denote the substitution a1a2... an.
Since there is no uniformity of usage along this line, the reader is frequently
obliged to determine the meaning from the context.