Central factor implies transitively normal
From Groupprops
This article gives the statement and possibly, proof, of an implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., central factor) must also satisfy the second subgroup property (i.e., transitively normal subgroup)
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Statement
Verbal statement
Any central factor of a group is a transitively normal subgroup. In other words, any normal subgroup of a central factor is a central factor.
Statement with symbols
Suppose is a central factor of a group
. Then,
is a transitively normal subgroup of
, i.e., if
is a normal subgroup of
, then
is a normal subgroup of
.
Proof
Given: A group , a central factor
of
, a normal subgroup
of
.
is an element of
and
is an element of
.
To prove: .
Proof: Let be the inner automorphism obtained as conjugation by
.
- There exists
such that the inner automorphism
by
, as an inner automorphism of
, is the restriction to
of
. In other words, for any
,
: This follows from the definition of central factor.
-
: This follows from the fact that
is normal in
.
-
: This follows immediately from the previous two steps.