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

~ 44]

CHARACTERISTIC SUBGROUPS

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it is clear that a characteristic subgroup of G cannot involve
any of its operators of order pal. All the operators of G whose
orders divide pal-a, where Al has any value from 1 to al-1,
constitute a characteristic subgroup of G. The characteristic subgroup CO of the preceding theorem corresponds to the
case when 1- = a-1. If all the invariants of G are equal,
there is only one characteristic subgroup of G, besides the
identity, which involves operators of order pB, but none of
higher order. It is also evident that the p'th powers of all
the operators of G constitute a characteristic subgroup of G.
If s is any independent generator of G, the conjugates of
s under the group of isomorphisms of G generate a group which
involves all the operators of G whose orders do not exceed
the order of s. In other words, if a characteristic subgroup involves an independent generator of an abelian group, it also
involves all the operators of this abelian group whose orders divide
the order of this independent generator. This theorem clearly
includes the theorem stated above, to the effect that a characteristic subgroup cannot involve any of the operators of
highest order contained in G. In the following paragraph we
shall establish a still more general theorem in case p> 2.
For a study of the special properties of the characteristic
subgroups it is convenient to let H1, H2,..., H represent the subgroups of G which are generated respectively by
a set of X1 independent generators of order pal, a set of X2
independent generators of order pa,..., a set of XA independent generators of order pa. Suppose that p>2 and that sl
is some operator of order p', ai> 6, which is contained in G.
If a,> - a,+1, and if si is the product of an operator of highest
order in H,+1 and an operator of order pa from H,, then the
conjugates of si under I generate a group which involves all
the operators of order p that are contained in G. In fact,
this group clearly involves an independent generator of H,+1
since p> 2, and it involves all the operators of order p8 in the
direct product of the subgroups H1,..., H,.
By means of the preceding theorems it is not difficult to
determine the characteristic subgroups of any given abelian