Normality satisfies intermediate subgroup condition: Difference between revisions

Latest revision as of 16:50, 24 February 2011

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 $H \leq K \leq G$ be groups such that $H ◃ G$ (viz., $H$ is normal in $G$ ). Then, $H$ is normal in $K$ .

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 $H$ is normal in $G$ and $K \leq G$ is any subgroup, then $H \cap K$ is normal in $K$
Inverse image condition	Normality satisfies inverse image condition	If $φ : K \to G$ is a homomorphism and $H$ is normal in $G$ , $φ^{- 1} (H)$ is normal in $K$
Image condition	Injective maps replaced by surjective maps	Normality satisfies image condition	If $φ : G \to K$ is surjective and $H$ is normal in $G$ , $φ (H)$ is normal in $K$
Upper join-closed subgroup property		Normality is upper join-closed	If $H \leq G$ and $K_{i}, i \in I$ are all subgroups of $G$ in which $H$ is normal, $H$ is normal in the join of the $K_{i}$ s

Related isomorphism theorems

Fourth isomorphism theorem (also called the lattice isomorphism theorem or correspondence theorem): This states that if $H$ is normal in $G$ , the quotient map $G \to G / H$ establishes a bijection between subgroups of $G$ containing $H$ (which is also a normal subgroup in each such subgroup) and subgroups of $G / H$ .
Third isomorphism theorem: This states that if $H \leq K \leq G$ and both $H, K$ are normal in $G$ , then $H$ is normal in $K$ , $K / H$ is normal in $G / H$ , and $G / K ≅ (G / H) / (K / H)$ .

General conditions to ensure intermediate subgroup condition

Intermediate subgroup condition for related properties

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

I-automorphism-invariance satisfies intermediate subalgebra condition: An I-automorphism in a variety of algebras is an automorphism expressible by a formula that is always guaranteed to yield automorphisms. In the variety of groups, the I-automorphisms are precisely the inner automorphisms.
Ideal property satisfies intermediate subalgebra condition: In any variety of algebras, an ideal of an algebra is also an ideal in every intermediate subalgebra containing it.
Ideal property satisfies intermediate subring condition in Lie rings: In a Lie ring, any ideal is also an ideal in every intermediate Lie subring.
Normality satisfies intermediate subloop condition: A normal subloop of an algebra loop is also normal in every intermediate subloop.

Proof

Hands-on proof

Given: $H \leq K \leq G$ such that $H ◃ G$

To prove: $H ◃ K$ : for any $g \in K$ , $g H g^{- 1} = H$ .

Proof: Pick any $g \in K$ . Since $K \leq G$ , $g \in G$ . Further, since $H$ is normal in $G$ and $g \in G$ , $g H g^{- 1} = H$ .

Proof in terms of inner automorphisms

This 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: $H \leq K \leq G$ , such that $H$ is invariant under all inner automorphisms of $G$ .

To prove: $H$ is invariant under all inner automorphisms of $K$ .

Proof: Suppose $σ$ is an inner automorphism of $K$ . Our goal is to show that $σ (H) \leq H$ .

Since $σ$ is inner in $K$ , there exists $g \in K$ such that $σ = c_{g}$ . In other words, $σ (x) = g x g^{- 1}$ for all $x \in H$ .
Since $K \leq G$ and $g \in K$ , we have $g \in G$ .
The map $c_{g} : x \mapsto g x g^{- 1}$ defines an inner automorphism $σ^{'}$ of the whole group $G$ , whose restriction to $K$ is $σ$ .
Since $H$ is normal in $G$ , $σ^{'} (H) \leq H$ .
Since the restriction of $σ^{'}$ to $K$ is $σ$ , and $H \leq K$ , we get $σ (H) \leq H$ .

Proof in terms of ideals

This 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 $H$ of a group $G$ is an ideal if for any expression $φ (u_{1}, u_{2}, \dots, u_{m}, t_{1}, t_{2}, \dots, t_{n})$ with the property that whenever all the $u_{i}$ are zero, the expression simplifies to zero, it is also true that whenever all the $u_{i}$ are in $H$ and the $t_{i}$ s are in $G$ , the expression yields a value in $G$ .

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 $G$ , an ideal $H$ of $G$ , a subgroup $K$ of $G$ containing $H$ .

To prove: $H$ is an ideal of $K$ . In other words, for any formula $φ (u_{1}, u_{2}, \dots, u_{m}, t_{1}, t_{2}, \dots, t_{n})$ that simplies to the identity element whenever the $u_{i}$ s are the identity element, we should have that the expression simplifies to a value inside $H$ whenever the $u_{i}$ are in $H$ and the $t_{i}$ are in $K$ .

Proof: Notice that since the $t_{i}$ are in $K$ , they are also in $G$ . Since we know that $H$ is an ideal in $G$ , we know by the property of $φ$ that $φ (u_{1}, u_{2}, \dots, u_{m}, t_{1}, t_{2}, \dots, t_{n}) \in H$ . This completes the proof.

Proof in terms of kernel of homomorphism

Given: A group $G$ , a subgroup $H$ of $G$ that is the kernel of a homomorphism $f : G \to L$ . A subgroup $K$ of $G$ containing $H$ .

To prove: $H$ is the kernel of a homomorphism originating from $K$ .

Proof: Let $i : K \to G$ be the inclusion map, and $g = f \circ i$ . In other words, $g : K \to L$ is the restriction of $f$ to $K$ . Then, $g$ is a homomorphism of groups (because it is a composite of two homomorphisms), and the kernel of $g$ is $H$ , completing the proof.

@@ Line 1: / Line 1: @@
 {{basic fact}}
-{{subgroup metaproperty satisfaction}}
+{{subgroup metaproperty satisfaction|
+property = normal subgroup|
+metaproperty = intermediate subgroup condition}}
+[[difficulty level::1| ]]
 ==Statement==
@@ Line 11: / Line 13: @@
 ===Statement with symbols===
-Let <math>H \le K \le G</math> be groups such that <math>H \triangleleft G</math> (viz <math>H</math> is [[normal subgroup|normal]] in <math>G</math>). Then, <math>H</math> is normal in <math>K</math>.
+Let <math>H \le K \le G</math> be groups such that <math>H \triangleleft G</math> (viz., <math>H</math> is [[normal subgroup|normal]] in <math>G</math>). Then, <math>H</math> is normal in <math>K</math>.
 ===Property-theoretic statement===
@@ Line 17: / Line 19: @@
 The [[subgroup property]] of being [[normal subgroup|normal]] satisfies the [[fact about::intermediate subgroup condition]].
-==Generalizations==
+==Related facts==
+===Related metaproperties satisfied by normality===
+Here are some related metaproperties that normality satisfies:
+{| class="sortable" border="1"
+! Metaproperty name !! Relation !! Proof of satisfaction !! Full statement
+|-
+| [[Transfer condition]] || Stronger than intermediate subgroup condition || [[Weaker than::Normality satisfies transfer condition]] || If <math>H</math> is normal in <math>G</math> and <math>K \le G</math> is any subgroup, then <math>H \cap K</math> is normal in <math>K</math>
+|-
+| [[Inverse image condition]] || [[Weaker than::Normality satisfies inverse image condition]] || If <math>\varphi:K \to G</math> is a homomorphism and <math>H</math> is normal in <math>G</math>, <math>\varphi^{-1}(H)</math> is normal in <math>K</math>
+|-
+| [[Image condition]] || Injective maps replaced by surjective maps || [[Normality satisfies image condition]] || If <math>\varphi:G \to K</math> is surjective and <math>H</math> is normal in <math>G</math>, <math>\varphi(H)</math> is normal in <math>K</math>
+|-
+| [[Upper join-closed subgroup property]] || || [[Normality is upper join-closed]] || If <math>H \le G</math> and <math>K_i, i \in I</math> are all subgroups of <math>G</math> in which <math>H</math> is normal, <math>H</math> is normal in the join of the <math>K_i</math>s
+|}
-===Stronger metaproperties satisfied by normality===
+===Related isomorphism theorems===
-* [[Normality satisfies transfer condition]]
+* [[Part of::Fourth isomorphism theorem]] (also called the lattice isomorphism theorem or correspondence theorem): This states that if <math>H</math> is normal in <math>G</math>, the quotient map <math>G \to G/H</math> establishes a bijection between subgroups of <math>G</math> containing <math>H</math> (which is also a normal subgroup in each such subgroup) and subgroups of <math>G/H</math>.
-* [[Normality satisfies inverse image condition]]
+* [[Part of::Third isomorphism theorem]]: This states that if <math>H \le K \le G</math> and both <math>H,K</math> are normal in <math>G</math>, then <math>H</math> is normal in <math>K</math>, <math>K/H</math> is normal in <math>G/H</math>, and <math>G/K \cong (G/H)/(K/H)</math>.
+===General conditions to ensure intermediate subgroup condition===
-===Weaker conditions to ensure intermediate subgroup condition===
+* [[Weaker than::Left-inner implies intermediate subgroup condition]]
+* [[Weaker than::Left-extensibility-stable implies intermediate subgroup condition]]
-* [[Left-inner implies intermediate subgroup condition]]
+===Intermediate subgroup condition for related properties===
-* [[Left-extensibility-stable implies intermediate subgroup condition]]
-==Related results==
+Here are some other properties that satisfy the intermediate subgroup condition:
-===Other subgroup properties satisfying intermediate subgroup condition===
+{| class="sortable" border="1"
+! 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]]
+|}
-* [[Central factor satisfies intermediate subgroup condition]]
+Here are some that don't:
-* [[Direct factor satisfies intermediate subgroup condition]]
+{| class="sortable" border="1"
+! Property !! Meaning !! Proof that it dissatisfies intermediate subgroup condition !! Relation with normality (in meaning and proof)
+|-
+| [[Characteristic subgroup]] || invariant under all [[automorphism]]s || [[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 [[endomorphism]]s || [[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===
+* [[I-automorphism-invariance satisfies intermediate subalgebra condition]]: An [[I-automorphism]] in a variety of algebras is an automorphism expressible by a formula that is ''always'' guaranteed to yield automorphisms. In the [[variety of groups]], [[inner automorphisms are I-automorphisms in the variety of groups|the I-automorphisms are precisely the inner automorphisms]].
+* [[Ideal property satisfies intermediate subalgebra condition]]: In any variety of algebras, an ideal of an algebra is also an ideal in every intermediate subalgebra containing it.
+* [[Ideal property satisfies intermediate subring condition in Lie rings]]: In a [[Lie ring]], any [[ideal of a Lie ring|ideal]] is also an ideal in every intermediate Lie subring.
+* [[Normality satisfies intermediate subloop condition]]: A [[normal subloop]] of an [[algebra loop]] is also normal in every intermediate subloop.
 ==Proof==
@@ Line 40: / Line 81: @@
 ===Hands-on proof===
-Given <math>H \le K \le G</math> such that <math>H \triangleleft G</math>, we need to show that <math>H \triangleleft K</math>. To prove this, it suffices to show that for any <math>g \in K</math>, <math>gHg^{-1} = H</math>.
+'''Given''': <math>H \le K \le G</math> such that <math>H \triangleleft G</math>
+'''To prove''': <math>H \triangleleft K</math>: for any <math>g \in K</math>, <math>gHg^{-1} = H</math>.
+'''Proof''': Pick any <math>g \in K</math>. Since <math>K \le G</math>, <math>g \in G</math>. Further, since <math>H</math> is normal in <math>G</math> and <math>g \in G</math>, <math>gHg^{-1} = H</math>.
+===Proof in terms of inner automorphisms===
+{{proof generalizes|[[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''': <math>H \le K \le G</math>, such that <math>H</math> is invariant under all inner automorphisms of <math>G</math>.
+'''To prove''': <math>H</math> is invariant under all inner automorphisms of <math>K</math>.
+'''Proof''': Suppose <math>\sigma</math> is an inner automorphism of <math>K</math>. Our goal is to show that <math>\sigma(H) \le H</math>.
+# Since <math>\sigma</math> is inner in <math>K</math>, there exists <math>g \in K</math> such that <math>\sigma = c_g</math>. In other words, <math>\sigma(x) = gxg^{-1}</math> for all <math>x \in H</math>.
+# Since <math>K \le G</math> and <math>g \in K</math>, we have <math>g \in G</math>.
+# The map <math>c_g: x \mapsto gxg^{-1}</math> defines an inner automorphism <math>\sigma'</math> of the ''whole'' group <math>G</math>, whose restriction to <math>K</math> is <math>\sigma</math>.
+# Since <math>H</math> is normal in <math>G</math>, <math>\sigma'(H) \le H</math>.
+# Since the restriction of <math>\sigma'</math> to <math>K</math> is <math>\sigma</math>, and <math>H \le K</math>, we get <math>\sigma(H) \le H</math>.
+===Proof in terms of ideals===
+{{proof generalizes|[[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 [[ideal in a variety with zero|ideal]]s in this variety are defined as follows: a subset <math>H</math> of a group <math>G</math> is an ideal if for any expression <math>\varphi(u_1, u_2, \dots, u_m, t_1, t_2, \dots, t_n)</math> with the property that whenever all the <math>u_i</math> are zero, the expression simplifies to zero, it is also true that whenever all the <math>u_i</math> are in <math>H</math> and the <math>t_i</math>s are in <math>G</math>, the expression yields a value in <math>G</math>.
+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 <math>G</math>, an ideal <math>H</math> of <math>G</math>, a subgroup <math>K</math> of <math>G</math> containing <math>H</math>.
-Pick any <math>g \in K</math>. Then, since <math>K \le G</math>, <math>g \in G</math>. But since <math>H</math> is normal in <math>G</math>, <math>gHg^{-1} = H</math>. This proves it.
+'''To prove''': <math>H</math> is an ideal of <math>K</math>. In other words, for any formula <math>\varphi(u_1, u_2, \dots, u_m, t_1, t_2, \dots, t_n)</math> that simplies to the identity element whenever the <math>u_i</math>s are the identity element, we should have that the expression simplifies to a value inside <math>H</math> whenever the <math>u_i</math> are in <math>H</math> and the <math>t_i</math> are in <math>K</math>.
-===Deeper insight leading to generalization===
+'''Proof''': Notice that since the <math>t_i</math> are in <math>K</math>, they are also in <math>G</math>. Since we know that <math>H</math> is an ideal in <math>G</math>, we know by the property of <math>\varphi</math> that <math>\varphi(u_1, u_2, \dots, u_m, t_1, t_2, \dots, t_n) \in H</math>. This completes the proof.
-We need to show that given any inner automorphism <math>\sigma</math> of <math>K</math>, <math>H</math> is invariant under <math>\sigma</math>.
+===Proof in terms of kernel of homomorphism===
-We know that given any inner automorphism of the ''whole group'' <math>G</math>, <math>H</math> is invariant under that. Thus, what we need to do is ''extend'' <math>\sigma</math> to an inner automorphism <math>\sigma'</math> of the whole of <math>G</math>. In other words, we need to show that any inner automorphism of a subgroup can be lifted to an inner automorphism of the whole group.
+'''Given''': A group <math>G</math>, a subgroup <math>H</math> of <math>G</math> that is the kernel of a homomorphism <math>f:G \to L</math>. A subgroup <math>K</math> of <math>G</math> containing <math>H</math>.
-This in turn follows easily from the fact that an inner automorphism is described via conjugation by an element of the subgroup, and conjugation by the same element also defines an inner automorphism on the whole group.
+'''To prove''': <math>H</math> is the kernel of a homomorphism originating from <math>K</math>.
-This leads to the generalizations mentioned above: [[left-inner implies intermediate subgroup condition|any left-inner subgroup property satisfies the intermediate subgroup condition]] and [[left-extensibility-stable implies intermediate subgroup condition|any left-extensibility-stable subgroup property satisfies the intermediate subgroup condition]].
+'''Proof''': Let <math>i:K \to G</math> be the inclusion map, and <math>g = f \circ i</math>. In other words, <math>g:K \to L</math> is the restriction of <math>f</math> to <math>K</math>. Then, <math>g</math> is a homomorphism of groups (because it is a composite of two homomorphisms), and the kernel of <math>g</math> is <math>H</math>, completing the proof.