J. Group Theory 9 (2006), 307–316
Journal of Group Theory
( de Gruyter 2006
The direct extension theorem
(Communicated by R. M. Guralnick)
Let H and K be ﬁnite groups and consider an extension 1 ! H ! G ! K ! 1. In
general the isomorphism class of G does not determine the extension. The main result
of this paper is it does in the special case that G G H Â K. Our theorem can be re-
stated as follows:
Theorem 1.1. Let G ¼ H Â K be a ﬁnite group. Suppose that H
is a normal subgroup
of G and assume that H
G H and G=H
G G=H. Then H
is a direct factor of G (i.e.
G ¼ H
for some complement K
The corresponding result for ﬁnitely generated modules over a noetherian com-
mutative ring is well known (see  or ). In particular, the result holds for ﬁnitely
generated abelian groups.
2 Preliminary results
In this section, G is any ﬁnite group, not necessarily the group that appears in the
2.1 Subgroups of a direct product G F H D K. We shall often use the following el-
ementary results without comment. We write ZðGÞ and G
for the center and derived
group of a group G.
Lemma 2.1. Let L be a subgroup of G, which we do not assume to be normal,
and G ¼ HK a decomposition of G into direct factors. Assume that H J L. Then
L ¼ HðL V KÞ. In particular, H is a direct factor of L.
Lemma 2.2. Let G ¼ HK be a decomposition of G into direct factors. Then G