Abnormal subgroup
From Encyclopedia of Mathematics
A subgroup of a group
such that
for any element
. Here
is the subgroup generated by
and its conjugate subgroup
. As an example of an abnormal subgroup of a finite group
one can take the normalizer (cf. Normalizer of a subset)
of any Sylow
-subgroup
, and even any maximal subgroup
which is not normal in
. In the theory of finite solvable groups (cf. Solvable group), where many important classes of subgroups are abnormal, use is made of the concept of a subabnormal subgroup
of a group
, which is defined by a series of subgroups
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where is abnormal in
,
.
References
[1] | B. Huppert, "Endliche Gruppen" , 1 , Springer (1967) |
Comments
Nowadays, is mostly defined as
. Instead of "solvable" also "soluble groupsoluble" is frequently used.
How to Cite This Entry:
Abnormal subgroup. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Abnormal_subgroup&oldid=33149
Abnormal subgroup. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Abnormal_subgroup&oldid=33149
This article was adapted from an original article by A.I. Kostrikin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article