IB 11
Perhaps this is a good place to digress to review some basic facts about
characteristic subgroups. (Some of this material also appears in the appen-
dix.) First, we recall the definition: a subgroup K C G is characteristic
in G if every automorphism of G maps K onto itself.
It is often difficult to find all automorphisms of a given group, and so the
definition of "characteristic" can be hard to apply directly, but nevertheless,
in many cases, it easy to establish that certain subgroups are characteristic.
For example, the center Z(G), the derived (or commutator) subgroup G',
and the intersection of all Sylow p-subgroups Op(G) are characteristic in
G. More generally, any subgroup that can be described unambiguously as
"the something" is characteristic. It is essential that the description using
the definite article be unambiguous, however. Given a subgroup H C G, for
example, we cannot conclude that the normalizer
'NG(H)
or the center
Z(JET)
is characteristic in G. Although these subgroups are described using "the",
the descriptions are not unambiguous because they depend on the choice
of H. We can say, however, that Z(G") is characteristic in G because it is
the center of the derived subgroup; it does not depend on any unspecified
subgroups.
A good way to see why "the something" subgroups must be characteris-
tic is to imagine two groups G\ and G2, with an isomorphism 9 : G\ G2.
Since isomorphisms preserve "group theoretic" properties, it should be clear
that 9 maps the center Z(G\) onto Z(G2), and indeed 9 maps each un-
ambiguously defined subgroup of G\ onto the corresponding subgroup of
G2. Now specialize to the case where G\ and G2 happen to be the same
group £?, so 9 is an automorphism of G. Since in the general case, we
know that 9(Z{G\)) = Z(C?2), we see that when G\ G = G?2, we have
0(Z(G)) = Z(G), and similarly, if we consider any "the something" sub-
group in place of the center.
Of course, characteristic subgroups are automatically normal. This
is because the definition of normality requires only that the subgroup be
mapped onto itself by inner automorphisms while characteristic subgroups
are mapped onto themselves by all automorphisms. We have seen that some
characteristic subgroups are easily recognized, and it follows that these sub-
groups are obviously and automatically normal. For example, the subgroup
Op(G) is normal in G for all primes p.
The fact that characteristic subgroups are normal remains true in an
even more general context. The following, which we presume is already
known to most readers of this book, is extremely useful. (This result also
appears in the appendix.)
1.10. Lemma. Let K C N C G, where G is a group, N is a normal
subgroup of G and K is a characteristic subgroup of N. Then K G.
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