Normality satisfies intermediate subgroup condition
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This article gives the statement, and possibly proof, of a basic fact in group theory.
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This article gives the statement, and possibly proof, of a subgroup property (i.e., normal subgroup) satisfying a subgroup metaproperty (i.e., intermediate subgroup condition)
View all subgroup metaproperty satisfactions | View all subgroup metaproperty dissatisfactions |Get help on looking up metaproperty (dis)satisfactions for subgroup properties
Get more facts about normal subgroup |Get facts that use property satisfaction of normal subgroup | Get facts that use property satisfaction of normal subgroup|Get more facts about intermediate subgroup condition
Statement
Verbal statement
If a subgroup is normal in the whole group, it is also normal in every intermediate subgroup of the group containing it.
Statement with symbols
Let be groups such that
(viz.,
is normal in
). Then,
is normal in
.
Property-theoretic statement
The subgroup property of being normal satisfies the Intermediate subgroup condition (?).
Related facts
Related metaproperties satisfied by normality
Here are some related metaproperties that normality satisfies:
Metaproperty name | Relation | Proof of satisfaction | Full statement |
---|---|---|---|
Transfer condition | Stronger than intermediate subgroup condition | Normality satisfies transfer condition | If ![]() ![]() ![]() ![]() ![]() |
Inverse image condition | Normality satisfies inverse image condition | If ![]() ![]() ![]() ![]() ![]() | |
Image condition | Injective maps replaced by surjective maps | Normality satisfies image condition | If ![]() ![]() ![]() ![]() ![]() |
Upper join-closed subgroup property | Normality is upper join-closed | If ![]() ![]() ![]() ![]() ![]() ![]() |
Related isomorphism theorems
- Fourth isomorphism theorem (also called the lattice isomorphism theorem or correspondence theorem): This states that if
is normal in
, the quotient map
establishes a bijection between subgroups of
containing
(which is also a normal subgroup in each such subgroup) and subgroups of
.
- Third isomorphism theorem: This states that if
and both
are normal in
, then
is normal in
,
is normal in
, and
.
General conditions to ensure intermediate subgroup condition
- Left-inner implies intermediate subgroup condition
- Left-extensibility-stable implies intermediate subgroup condition
Here are some other properties that satisfy the intermediate subgroup condition:
Property | Meaning | Proof that it satisfies intermediate subgroup condition | Relation with normality (in meaning and proof) |
---|---|---|---|
Central factor | every inner automorphism of whole group restricts to inner automorphism of subgroup | Central factor satisfies intermediate subgroup condition | both are examples of left-inner implies intermediate subgroup condition |
Direct factor | factor in an internal direct product | Direct factor satisfies intermediate subgroup condition | |
Subnormal subgroup | finite chain from subgroup to group, each normal in next | Subnormality satisfies intermediate subgroup condition | This actually follows from the stronger fact that normality satisfies transfer condition and transfer condition is composition-closed |
Here are some that don't:
Property | Meaning | Proof that it dissatisfies intermediate subgroup condition | Relation with normality (in meaning and proof) | |
---|---|---|---|---|
Characteristic subgroup | invariant under all automorphisms | Characteristicity does not satisfy intermediate subgroup condition | The proof fails because automorphisms cannot always be extended to bigger groups (see extensible automorphisms problem) | |
Full invariance does not satisfy intermediate subgroup condition | invariant under all endomorphisms | Full invariance does not satisfy intermediate subgroup condition | The proof fails because endomorphisms cannot always be extended to bigger groups | ]
Analogues in other algebraic structures
ProofHands-on proofGiven: To prove: Proof: Pick any Proof in terms of inner automorphismsThis proof method generalizes to the following results: I-automorphism-invariance satisfies intermediate subalgebra condition over arbitrary varieties of algebras, left-inner implies intermediate subgroup condition, and left-extensibility-stable implies intermediate subgroup condition The key idea here is that since inner automorphisms can be expressed by a formula that is guaranteed to yield an automorphism, any inner automorphism of a smaller subgroup extends to an inner automorphism of a bigger subgroup. Given: To prove: Proof: Suppose
Proof in terms of idealsThis proof method generalizes to the following results: ideal property satisfies intermediate subalgebra condition over arbitrary varieties of algebras with zero. The key idea here is to view the variety of groups as a variety with zero, i.e., a variety of algebras with a distinguished constant operation -- in this case, the identity element. The ideals in this variety are defined as follows: a subset It turns out that the ideals in the variety of groups are precisely the same as the normal subgroups (this is a consequence of the proof that the variety of groups is ideal-determined). We thus give the proof in terms of ideals in the variety of groups, assuming the equivalence. Given: A group To prove: Proof: Notice that since the Proof in terms of kernel of homomorphismGiven: A group To prove: Proof: Let |