Minimal normal implies characteristically simple: Difference between revisions

Latest revision as of 22:19, 23 September 2008

This article describes a fact or result that is not basic but it still well-established and standard. The fact may involve terms that are themselves non-basic
View other semi-basic facts in group theory
VIEW FACTS USING THIS: directly | directly or indirectly, upto two steps | directly or indirectly, upto three steps|
VIEW: Survey articles about this

Template:Subgroup-to-group property implication

This fact is an application of the following pivotal fact/result/idea: characteristic of normal implies normal
View other applications of characteristic of normal implies normal OR Read a survey article on applying characteristic of normal implies normal

Statement

Verbal statement

Any group that occurs as a minimal normal subgroup of some group is characteristically simple.

Definitions

Minimal normal subgroup

PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]

Characteristically simple group

PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]

Related facts

Similar facts for specific kinds of groups

Converse

Characteristically simple equals left-realization of minimal normal: Every characteristically simple group can be embedded as a minimal normal subgroup inside some group (in fact, inside its holomorph).

Facts used

Characteristic of normal implies normal: Every characteristic subgroup of a normal subgroup is normal in the whole group.

Proof

Hands-on proof

Given: $H$ is a minimal normal subgroup of a group $G$ .

To prove: $H$ is characteristically simple.

Proof: Suppose $K$ is a characteristic subgroup of $H$ . Then, by fact (1), $K$ is normal in $G$ . But since $H$ was minimal normal, either $K=H$ or $K$ is trivial. Thus, every characteristic subgroup of $H$ is either the whole of $H$ or is trivial.

@@ Line 1: / Line 1: @@
+{{semibasic fact}}
 {{subgroup-to-group property implication}}
-{{semibasic fact}}
-{{applicationof|characteristic of normal implies normal}}
+{{application of|characteristic of normal implies normal}}
 ==Statement==
@@ Line 8: / Line 9: @@
 Any group that occurs as a [[minimal normal subgroup]] of some group is [[characteristically simple group|characteristically simple]].
-We shall also prove a converse: every characteristically simple group can be embedded as a minimal normal subgroup inside some group (in fact, inside its [[holomorph]]).
 ==Definitions==
@@ Line 20: / Line 20: @@
 {{fillin}}
-==Results used==
+==Related facts==
+===Similar facts for specific kinds of groups===
-The main result used in this proof is that characteristicity is the [[left transiter]] for normality, viz every characteristic subgroup of a normal subgroup is normal
+* [[Minimal normal implies additive group of a field in solvable]]
-.
+* [[Minimal normal implies elementary Abelian in finite solvable]]
-==Proof==
+* [[Minimal normal implies contained in Omega-1 of center for nilpotent p-group]]
+* [[Minimal normal implies central in nilpotent]]
+===Converse===
+* [[Characteristically simple equals left-realization of minimal normal]]: Every characteristically simple group can be embedded as a minimal normal subgroup inside some group (in fact, inside its [[holomorph]]).
-===Hands-on proof===
+==Facts used==
-Let <math>H</math> be a minimal normal subgroup of a group <math>G</math>. We want to show that <math>H</math> is characteristically simple.
+# [[Characteristic of normal implies normal]]: Every characteristic subgroup of a normal subgroup is normal in the whole group.
-Suppose <math>K</math> is a [[characteristic subgroup]] of <math>H</math>. Then, since every [[characteristic of normal implies normal|characteristic subgroup of a normal subgroup is normal]], <math>K</math> is normal in <math>G</math>. But since <math>H</math> was ''minimal'' normal, either <math>K = H</math> or <math>K</math> is trivial. Thus, every characteristic subgroup of <math>H</math> is either the whole of <math>H</math> or is trivial.
+==Proof==
-==Proof of converse==
+===Hands-on proof===
-===Explicit construction===
+'''Given''': <math>H</math> is a minimal normal subgroup of a group <math>G</math>.
-We can check that any characteristically simple group is minimal normal in its holomorph. This follows from the fact that every automorphism of the group lifts to an inner automorphism in the holomorph.
+'''To prove''': <math>H</math> is characteristically simple.
-{{fillin}}
+'''Proof''': Suppose <math>K</math> is a [[characteristic subgroup]] of <math>H</math>. Then, by fact (1), <math>K</math> is normal in <math>G</math>. But since <math>H</math> was ''minimal'' normal, either <math>K = H</math> or <math>K</math> is trivial. Thus, every characteristic subgroup of <math>H</math> is either the whole of <math>H</math> or is trivial.