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

~ 24]

PROPERTIES OF TRANSFORMS

57

conditions that there is only one group of order g are that g
is not divisible by the square of a prime number and that none
of its prime factors is a divisor of the number obtained by
diminishing by unity another such factor.
EXERCISES
1. Prove that the 8 natural numbers which are less than 15 and prime
to 15 constitute a group with respect to multiplication (mod 15), which
is not simply isomorphic with the group formed similarly by the numbers
which are less than 24 and prime to 24.
2. The smallest group of multiplication which involves the two
matrices
1 0 0 1
0-1 1 0
is of order 8. Is this group simply isomorphic with either of the groups
of Exercise 1? Find the six other matrices of this group.
3. To which of the three groups of the preceding exercises is the
group of movements of the square simply isomorphic? To which is the
group formed by the 8 numbers less than 20 and prime to 20, with respect
to multiplication (mod 20), simply isomorphic?
4. Find two groups whose product is the cyclic group of order 36,
and determine the number of elements of each order in this group.
5. Including the identity there are five complete sets of conjugate
elements in the group of movements of the square. Determine the elements of each of these sets.
6. Do the numbers 2, 4, 6, 8 form a group with respect to multiplication (mod 10)? If so, is this group simply isomorphic with the group
formed by 1, -1, /-l, - /-1? Which of the four numbers 2, 4, 6, 8
corresponds to the identity?
7. If s6 is of order 2 find two substitutions of orders 12 and 4 respectively which may be used for s.
8. Give the orders of all the possible cyclic groups having only two
distinct generators.
24. Properties of Transforms. In ~ 9 we considered the
transform of a substitution. As the concept of transforming is
very useful we shall develop the properties of this operation
more fully at this place. Suppose that
s- ts = ".