Conjugacy-closedness is transitive
From Groupprops
This article gives the statement, and possibly proof, of a subgroup property (i.e., conjugacy-closed subgroup) satisfying a subgroup metaproperty (i.e., transitive subgroup property)
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Statement
Statement with symbols
Suppose are groups such that
is conjugacy-closed in
and
is conjugacy-closed in
.
Related facts
Proof
Hands-on proof
Given: Groups such that
is conjugacy-closed in
and
is conjugacy-closed in
.
To prove: is conjugacy-closed in
.
Proof: We need to show that if are conjugate in
, then they are conjugate in
. First, observe that since
,
are elements of
conjugate in
. Since
is conjugacy-closed in
,
are conjugate in
.
Thus, are elements of
that are conjugate in
. Since
is conjugacy-closed in
, the elements
are conjugate in
.