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

~ 44]

CHARACTERISTIC SUBGROUPS

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of order 2aopllp2't... (pl, P2,... being distinct odd
primes) be conformal with at least one non-abelian group are:
1~ at least one of its subgroups of orders 2a0, pIa, p2a02,... is
non-cyclic; 20 if the order pfa, of this subgroup is odd, then
ao> 2; if the order is even (2a~), then the subgroup must involve
operators of order 4 and ao>3. Since any number of these
factors may be non-abelian, there cannot be an upper limit
to the number of non-abelian groups which can be conformal
with an arbitrary abelian group. This fact may be seen in
many other ways.
EXERCISES
1. Let s be of order 16 and let t represent an operator of order 2 such
that tst=s9; prove that (s, t) is conformal with the abelian group whose
invariants are 16, 2.
2. Find the group of isomorphisms of the abelian group whose invariants are 8, 2 and determine its invariant operators.
3. If p is an odd number, there is at least one non-abelian group of
order ptm, m>2, which is conformal with the abelian group of type (1,
1,... to m units). The number of such possible groups increases with
m and has no upper limit.
4. Any operator of order pa in any abelian group whatever can be
used as an independent generator provided its p -lth power is not
included in a cyclic subgroup of order p +1.
5. Every possible group of finite order is a subgroup of the group of
isomorphisms of an abelian group of order 2m and of type (1, 1, 1,...).
Suggestions: Observe that this group of isomorphisms contains a
subgroup which is simply isomorphic with the symmetric group of degree m.
44. Characteristic Subgroups of an Abelian Group. In ~ 29
a characteristic subgroup was defined as a subgroup which corresponds to itself in every possible automorphism of the group.
An operator which corresponds to itself in every possible
automorphism of the group is likewise called a characteristic
operator. It is clear that every characteristic subgroup is also
an invariant subgroup, and that every characteristic operator
is also an invariant operator; but invariant subgroups and
invariant operators are not necessarily characteristic. The
Sylow subgroups of an abelian group whose order is not a
power of a single prime are evidently characteristic subgroups.