Normal versus permutable - Revision history

Vipul: /* Upper join-closedness */

2009-02-25T15:34:47Z

Upper join-closedness

← Older revision		Revision as of 15:34, 25 February 2009
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	==Upper join-closedness==		==Upper join-closedness==

	{{further\|[[Normality is upper join-closed]], [[Permutability is upper join-closed]]}}		{{further\|[[Normality is upper join-closed]], [[Permutability is not upper join-closed]]}}

	If <math>H</math> is normal in a collection of intermediate subgroups <math>K_i</math> of a group <math>G</math>, then <math>H</math> is also normal in the join of the <math>K_i</math>s. ~~Similarly, if~~ <math>H</math> is permutable in ~~a collection of~~ intermediate subgroups <math>~~K_i~~</math> of <math>G</math>, <math>H</math> ~~is also~~ permutable in ~~their~~ join.		If <math>H</math> is normal in a collection of intermediate subgroups <math>K_i</math> of a group <math>G</math>, then <math>H</math> is also normal in the join of the <math>K_i</math>s.

			The corresponding statement is not true for permutability. If <math>H</math> is permutable in intermediate subgroups <math>K_1</math> and <math>K_2</math> of <math>G</math>, <math>H</math> need not be permutable in the join <math>\langle K_1, K_2 \rangle</math>.

	==Corresponding notions of simplicity==		==Corresponding notions of simplicity==

Vipul: /* Some operators */

2009-02-25T01:25:09Z

Some operators

← Older revision		Revision as of 01:25, 25 February 2009
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	==Some operators==		==Some operators==

	===Hereditarily operator===		===Hereditarily operator and Hamiltonian operator===

	{{further\|[[Hereditarily normal subgroup]], [[Hereditarily permutable subgroup]], [[Dedekind group]], [[PH-group]]}}		{{further\|[[Hereditarily normal subgroup]], [[Hereditarily permutable subgroup]], [[Dedekind group]], [[PH-group]]}}

Vipul at 01:24, 25 February 2009

2009-02-25T01:24:25Z

@@ Line 51: / Line 51: @@
 It is clear from the definition that if <math>H</math> is normal in <math>G</math>, then <math>H</math> is also normal in any intermediate subgroup. It is further clear that, for any subgroup <math>K \le G</math>, <math>H \cap K</math> is normal in <math>K</math>. Finally, the inverse image of a normal subgroup under any homomorphism is also normal.
-===Permutability satisfies intermediate subgroup condition, trasnfer condition, inverse image condition===
+===Permutability satisfies intermediate subgroup condition, transfer condition, inverse image condition===
 {{further|[[Permutability satisfies intermediate subgroup condition]], [[Permutability satisfies transfer condition]], [[Normality satisfies inverse image condition]]}}
@@ Line 65: / Line 65: @@
 A normal subgroup of a normal subgroup need not be normal. Similarly, a permutable subgroup of a permutable subgroup need not be permutable.
-===Normality has left and right transiters===
+===Left transiter for normality===
-The ''lack of transitivity'' of normality can be remedied somewhat. In particular:
+{{further|[[Characteristic of normal implies normal]], [[Left transiter of normal is characteristic]]}}
-# [[Characteristic of normal implies normal|Any characteristic subgroup of a normal subgroup is normal]]: Thus, for instance, the [[center]] and the [[commutator subgroup]] of a normal subgroup are normal.
+If <math>H</math> is a characteristic subgroup of <math>K</math> and <math>K</math> is a normal subgroup of <math>G</math>, then <math>H</math> is normal in <math>G</math> as well.
-===Permutability has no clearly identifiable left or right transiters===
+In fact, if <math>H \le K</math> is a subgroup such that whenever <math>K</math> is normal in some bigger group <math>G</math>, so is <math>H</math>, we must have that <math>H</math> is characteristic in <math>K</math>.
 ==Effect of joins and intersections==
@@ Line 83: / Line 89: @@
 ===Permutability is closed under joins but not under intersections===
-{{further|[[Permutability is strongly join-closed]], [[Permutability is not intersection-closed]]}}
+{{further|[[Permutability is strongly join-closed]], [[Permutability is not finite-intersection-closed]]}}
 ==Upper join-closedness==
 ==Corresponding notions of simplicity==
-{{fillin}}
+* [[Simple group]] is a group that has no proper nontrivial normal subgroups.

Vipul at 23:32, 7 August 2008

2008-08-07T23:32:44Z

← Older revision		Revision as of 23:32, 7 August 2008
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	{{comparison survey article\|~~normality~~\|~~permutability~~}}		{{subgroup property comparison survey article\|
			first = normal subgroup\|
			second = permutable subgroup}}

	==Introduction==		==Introduction==

Vipul: New page: {{comparison survey article|normality|permutability}} ==Introduction== This article is about the relation, the similarity and contrast, between the well-known subgroup property of [[norm...

2008-06-12T13:01:50Z

New page: {{comparison survey article|normality|permutability}} ==Introduction== This article is about the relation, the similarity and contrast, between the well-known subgroup property of [[norm...

New page

{{comparison survey article|normality|permutability}}

==Introduction==

This article is about the relation, the similarity and contrast, between the well-known subgroup property of [[normality]] and the somewhat more obscure subgroup property called [[permutability]]: permuting with every subgroup.

==Definitions==

===Normal subgroup===

{{further|[[Normal subgroup]]}}

A subgroup <math>H</math> of a group <math>G</math> is termed normal if for every <math>g \in G</math>:

<math>Hg = gH</math>

In other words, its left cosets equal its right cosets.

===Permutable subgroup===

{{further|[[Permutable subgroup]]}}

A subgroup <math>H</math> of a group <math>G</math> is termed '''permutable''' or '''quasinormal''' if for every subgroup <math>K</math> of <math>G</math>:

<math>HK = KH</math>

In other words, <math>H</math> and <math>K</math> are [[permuting subgroups]].

It actually suffices to check the condition only for all ''cyclic'' subgroups <math>K</math> of <math>G</math>.

==Implication relations==

===Every normal subgroup is permutable===

{{further|[[Normal implies permutable]]}}

Normality requires the subgroup to commute with every ''element''. Hence, it commutes with every subset, and in particular, with every subgroup.

===Every permutable subgroup need not be normal===

{{further|[[Permutable not implies normal]]}}

==Similar behavior with respect to intermediate subgroups==

===Normality satisfies intermediate subgroup condition, transfer condition, inverse image condition===

{{further|[[Normality satisfies intermediate subgroup condition]], [[Normality satisfies transfer condition]], [[Normality satisfies inverse image condition]]}}

It is clear from the definition that if <math>H</math> is normal in <math>G</math>, then <math>H</math> is also normal in any intermediate subgroup. It is further clear that, for any subgroup <math>K \le G</math>, <math>H \cap K</math> is normal in <math>K</math>. Finally, the inverse image of a normal subgroup under any homomorphism is also normal.

===Permutability satisfies intermediate subgroup condition, trasnfer condition, inverse image condition===

{{further|[[Permutability satisfies intermediate subgroup condition]], [[Permutability satisfies transfer condition]], [[Normality satisfies inverse image condition]]}}

If <math>H</math> is permutable in <math>G</math>, then <math>H</math> is also permutable in any intermediate subgroup. Further, it is true that for any <math>K \le G</math>, <math>H \cap K</math> is permutable in <math>K</math>. Finally, the inverse image of a permutable subgroup under any homomorphism is permutable.

==Transitivity and transiters==

===Neither property is transitive===

{{further|[[Normality is not transitive]], [[Permutability is not transitive]]}}

A normal subgroup of a normal subgroup need not be normal. Similarly, a permutable subgroup of a permutable subgroup need not be permutable.

===Normality has left and right transiters===

The ''lack of transitivity'' of normality can be remedied somewhat. In particular:

# [[Characteristic of normal implies normal|Any characteristic subgroup of a normal subgroup is normal]]: Thus, for instance, the [[center]] and the [[commutator subgroup]] of a normal subgroup are normal.
# Any normal subgroup of a so-called [[transitively normal subgroup]], is normal. In particular, any [[central subgroup]], [[hereditarily normal subgroup]], [[direct factor]] or [[central factor]] is transitively normal.

===Permutability has no clearly identifiable left or right transiters===

{{fillin}}
==Effect of joins and intersections==

===Normality is closed under both===

{{further|[[Normality is strongly join-closed]], [[normality is strongly intersection-closed]]}}

===Permutability is closed under joins but not under intersections===

{{further|[[Permutability is strongly join-closed]], [[Permutability is not intersection-closed]]}}
==Upper join-closedness==

==Corresponding notions of simplicity==

{{fillin}}