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.