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

ABELIAN GROUPS

[CH. IV

If g is written in the form g=2aoplap2"... (pl, p2,.
being different odd prime numbers), G is the direct product
of its subgroups of orders 20a, p', p 2a2,..., and its group
of isomorphisms I is evidently the direct product of the groups
of isomorphisms of these subgroups. Since the group of isomorphisms of a cyclic group whose order is a power of an odd
prime number is cyclic, it follows from the above that I is the
direct product of the cyclic groups of orders pjl1-l(pi-1),
2a20-(P2-1),..., when ao=0 or 1. When ao>l, we have
to add a group of order 2 and a cyclic group of order 2a0-2 to
these factor groups in order to obtain I, since there are numbers
which belong to the exponent 2ao-2 (mod 2"), but none which
belong to a higher exponent.*
Since I is the direct product of groups of even orders, the
order of I is always even when g>2. It can clearly be any
even number of the form 2~plp202... (p- 1)(p2- )...
The smallest two natural numbers which are not of this form
are 14 and 26;t hence these numbers are the lowest orders of
groups that cannot be groups of isomorphisms of any cyclic
group whatever. It is evident that the highest prime factor
of the order of I can not exceed the highest prime factor of g.
EXERCISES
1. Determine the invariants of the group formed by the 40= -(100)
totitives of 100 (mod 100).
2. The number of invariants in the group of the totitives of m (mod
m) is equal to the number of the distinct odd prime factors of m whenever m is either odd or double an odd number. It is equal to the number of distinct prime factors of m, whenever m is divisible by 4 but not
by 8; when m is divisible by 8, the number of these invariants is one
more than the number of the distinct prime factors of m. Cf. Weber,
Lehrbuch der Algebra, vol. 2, 1896, p. 59.
3. Find the three possible cyclic groups whose group of isomorphisms
has the invariants 6, 2, 2.
4. If the operators of order 2 in the group of isomorphisms of the cyclic
group of order m generate a group of order 2', what is the maximum number of the distinct primes which divide m? What is the minimum number
of such divisors of m?
* Cf. H. Weber, Lehrbuch der Algebra, 2d ed., vol. 2, 1899, D. 64.
t Lucas, Th6orie des nombres, 1891, p. 394.