Normal equals potentially characteristic
From Groupprops
This article gives a proof/explanation of the equivalence of multiple definitions for the term normal subgroup
View a complete list of pages giving proofs of equivalence of definitions
Contents
Statement
The following are equivalent for a subgroup of a group
:
-
is a normal subgroup of
.
-
is a potentially characteristic subgroup of
in the following sense: there exists a group
containing
such that
is a characteristic subgroup of
.
Related facts
Stronger facts
- Finite NPC theorem
- Finite NIPC theorem
- Fact about amalgam-characteristic subgroups: finite normal implies amalgam-characteristic, periodic normal implies amalgam-characteristic, central implies amalgam-characteristic
Facts used
- Characteristicity is centralizer-closed
- Characteristic implies normal
- Normality satisfies intermediate subgroup condition
Proof
Proof of (1) implies (2) (hard direction)
Given: A group , a normal subgroup
of
.
To prove: There exists a group containing
such that
is characteristic in
.
Proof:
Step no. | Assertion/construction | Facts used | Given data used | Previous steps used | Explanation |
---|---|---|---|---|---|
1 | Let ![]() ![]() |
Note that such a group exists. For instance, we can take the finitary alternating group on any set of cardinality strictly bigger than that of ![]() | |||
2 | Let ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Step (1) | |||
3 | Any homomorphism from ![]() ![]() |
Steps (1), (2) | By definition, ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
4 | ![]() ![]() |
Steps (2), (3) | Under any automorphism of ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
5 | The centralizer of ![]() ![]() ![]() |
Steps (1), (2) | By definition of the wreath product action, ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
6 | The centralizer of ![]() ![]() ![]() |
Steps (2), (5) | Step (5) already shows that ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
7 | ![]() ![]() |
Fact (1) | Steps (4), (6) | Step-fact combination direct. |
This proof uses a tabular format for presentation. Provide feedback on tabular proof formats in a survey (opens in new window/tab) | Learn more about tabular proof formats|View all pages on facts with proofs in tabular format
Proof of (2) implies (1) (easy direction)
Given: A group , a subgroup
of
, a group
containing
such that
is characteristic in
.
To prove: is normal in
.
Proof:
Step no. | Assertion/construction | Facts used | Given data used | Previous steps used | Explanation |
---|---|---|---|---|---|
1 | ![]() ![]() |
Fact (2) | ![]() ![]() |
-- | Given-fact-combination direct. |
2 | ![]() ![]() |
Fact (3) | ![]() |
Step (1) | Given-step-fact combination direct. |