Weakly closed implies conjugation-invariantly relatively normal in finite group
From Groupprops
Template:Subofsubgroup property implication in
Statement
Suppose are groups such that
is a Weakly closed subgroup (?) of
relative to
. Then,
is a Conjugation-invariantly relatively normal subgroup (?) of
relative to
, viz.,
is normal in every conjugate of
in
containing it.
Facts used
Proof
Given: Groups such that
is weakly closed in
with respect to
.
To prove: If is such that
, then
is normal in
.
Proof:
-
: Since
, we have
. Now, since
is weakly closed in
, we get that
.
-
: Since
is finite, and conjugation by
is an automorphism, the sizes of
and
are the same. This, along with the previous step, yields
.
-
: This follows from the previous step by conjugating both sides by
.
-
is weakly closed in
: Since
is weakly closed in
, and conjugation by
is an automorphism,
is weakly closed in
.
by the previos step, so
is weakly closed in
.
-
is normal in
: This follows from the previous step, and fact (1).