Contranormality is upper join-closed
From Groupprops
This article gives the statement, and possibly proof, of a subgroup property (i.e., contranormal subgroup) satisfying a subgroup metaproperty (i.e., upper join-closed subgroup property)
View all subgroup metaproperty satisfactions | View all subgroup metaproperty dissatisfactions |Get help on looking up metaproperty (dis)satisfactions for subgroup properties
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Contents
Statement
Statement with symbols
Suppose is a subgroup, and
, is an indexed family of subgroups with
for each
. Then, if
is contranormal in each
,
is also contranormal in the [join of subgroups|join]] of the
s.
Definitions used
Contranormal subgroup
Further information: contranormal subgroup
is a contranormal subgroup if for any
containing
such that
is normal in
,
.
Related facts
Stronger facts
Applications
Facts used
- Normality satisfies transfer condition: If
is a normal subgroup, and
, then
is normal in
.
Proof
Given: , family of subgroups
with
, and
contranormal in each
.
To prove: is normal in the join of all the
s.
Proof: Suppose is a normal subgroup of the join of the
s, containing
. Then, for each
,
is a subgroup of
containing
, and by fact (1), it is normal in
. Since
is contranormal in
,
for each
, so
for each
. Thus,
must equal the join of the
s.