Normality is not transitive: Difference between revisions

Revision as of 10:50, 20 June 2008

Statement

Verbal statement

A Normal subgroup (?) of a normal subgroup need not be normal.

Symbolic statement

There can be a situation where $H$ is a normal subgroup of $K$ and $K$ is a normal subgroup of $G$ but $H$ is not a normal subgroup of $G$ .

Property-theoretic statement

Normality is not a transitive subgroup property.

Partial truth

Transitivity-forcing operator

A group in which every normal subgroup of a normal subgroup is normal is termed a T-group.

Left transiter

While normality is not transitive, it is still true that every characteristic subgroup of a normal subgroup is normal. This is property-theoretically because characteristicity is the left transiter of normality. For full proof, refer: Characteristic of normal implies normal, Left transiter of normal is characteristic

Right transiter

While normality is not transitive, every normal subgroup of a transitively normal subgroup is normal. Being transitively normal is the right transiter of being normal. Properties like being a direct factor, being a central subgroup, and being a central factor imply being transitively normal.

Subnormality

The lack of transitivity of normality can also be remedied by defining the notion of subnormal subgroup. Subnormality is the weakest transitive subgroup property implied by normality. More explicitly, a subgroup $H$ is subnormal in a group $G$ , if we can find a chain of subgroups going up from $H$ to $G$ , with each subgroup normal in its successor.

A special case of this is the notion of 2-subnormal subgroup, which is a normal subgroup of a normal subgroup.

Examples

Generic example

A natural example is as follows. Take any nontrivial group $G$ , and consider the square, $K = G \times G$ (the external direct product of $G$ with itself). Now, consider the external semidirect product of this group with the group $Z / 2 Z$ (the cyclic group of two elements) acting via the exchange automorphism (the automorphism that exchanges the coordinates. Call the big group $L$ . Let $G_{1}, G_{2}$ be the copies of $G$ embedded in $K$ as the first and second coordinate. Then $G_{1} ◃ K ◃ L$ but $G ◃̸ L$ , because the exchange automorphism is inner in $L$ , and sends $G_{1}$ to $G_{2}$ .

Specific realizations of this generic example

The smallest case of this yields $G_{1}$ a group of order two, and $L$ a group of order eight. In fact, here $L$ is the dihedral group of order eight and $G$ is a cyclic group of order two, with $G_{1}$ and $G_{2}$ being subgroups of order two generated by reflections.
If $G$ is a group of prime order $p$ where $p$ is odd, then $L$ is an extraspecial group of order $p^{3}$ , and exponent $p$ . In other words, $L$ is the group of unitary upper triangular matrices of order three over the prime field.

References

Textbook references

Groups and representations by Jonathan Lazare Alperin and Rowen B. Bell, ISBN 0387945261, ^{More info}, Page 6 (first mention), Page 8 (counterexample), Page 17 (further explanation)
Abstract Algebra by David S. Dummit and Richard M. Foote, 10-digit ISBN 0471433349, 13-digit ISBN 978-0471433347, ^{More info}, Page 91, Page 135
An Introduction to Abstract Algebra by Derek J. S. Robinson, ISBN 3110175444^{More info}, Page 66
A Course in the Theory of Groups by Derek J. S. Robinson, ISBN 0387944613, ^{More info}, Page 17 (Exercise 1.3.15), Page 28, Page 63
Algebra by Michael Artin, ISBN 0130047635, 13-digit ISBN 978-0130047632, ^{More info}, Page 236, Exercise 4, Miscellaneous Problems (Chapter 6) (starred problem)

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@@ Line 3: / Line 3: @@
 ===Verbal statement===
-A [[normal subgroup]] of a [[normal subgroup]] need not be normal.
+A [[fact about::normal subgroup]] of a [[normal subgroup]] need not be normal.
 ===Symbolic statement===
@@ Line 12: / Line 12: @@
 Normality is ''not'' a [[transitive subgroup property]].
 ==Partial truth==
@@ Line 20: / Line 21: @@
 ===Left transiter===
-While normality is not transitive, it is still true that every [[characteristic subgroup]] of a normal subgroup is normal. This is property-theoretically because characteristicity is the [[left transiter]] of normality. {{proofat|[[Left transiter of normal is characteristic]]}}
+While normality is not transitive, it is still true that every [[characteristic subgroup]] of a normal subgroup is normal. This is property-theoretically because characteristicity is the [[left transiter]] of normality. {{proofat|[[Characteristic of normal implies normal]], [[Left transiter of normal is characteristic]]}}
 ===Right transiter===
-While normality is not transitive, every normal subgroup of a [[transitively normal subgroup]] is normal. Being transitively normal is the [[right transiter]] of being normal. Properties like being a [[direct factor]] and being a [[central factor]] imply normality.
+While normality is not transitive, every normal subgroup of a [[transitively normal subgroup]] is normal. Being transitively normal is the [[right transiter]] of being normal. Properties like being a [[direct factor]], being a [[central subgroup]], and being a [[central factor]] imply being transitively normal.
+===Subnormality===
+The lack of transitivity of normality can also be remedied by defining the notion of [[subnormal subgroup]]. Subnormality is the weakest transitive subgroup property implied by normality. More explicitly, a subgroup <math>H</math> is subnormal in a group <math>G</math>, if we can find a chain of subgroups going up from <math>H</math> to <math>G</math>, with each subgroup normal in its successor.
+A special case of this is the notion of [[2-subnormal subgroup]], which is a normal subgroup of a normal subgroup.
-==Example==
+==Examples==
+===Generic example===
 A ''natural'' example is as follows. Take any nontrivial group <math>G</math>, and consider the square, <math>K = G \times G</math> (the [[external direct product]] of <math>G</math> with itself). Now, consider the [[external semidirect product]] of this group with the group <math>\mathbb{Z}/2\mathbb{Z}</math> (the [[cyclic group:Z2|cyclic group of two elements]]) acting via the exchange automorphism (the automorphism that exchanges the coordinates. Call the big group <math>L</math>. Let <math>G_1, G_2</math> be the copies of <math>G</math> embedded in <math>K</math> as the first and second coordinate. Then <math>G_1 \triangleleft K \triangleleft L</math> but <math>G \not \triangleleft L</math>, because the exchange automorphism is inner in <math>L</math>, and sends <math>G_1</math> to <math>G_2</math>.
-The smallest case of this yields <math>G_1</math> a group of order two, and <math>L</math> a group of order eight. In fact, here <math>L</math> is the [[dihedral group of order eight]] and <math>G</math> is the [[cyclic group:Z2|cyclic group of order two]].
+===Specific realizations of this generic example===
+# The smallest case of this yields <math>G_1</math> a group of order two, and <math>L</math> a group of order eight. In fact, here <math>L</math> is the [[dihedral group of order eight]] and <math>G</math> is a [[cyclic group:Z2|cyclic group of order two]], with <math>G_1</math> and <math>G_2</math> being subgroups of order two generated by reflections.
+# If <math>G</math> is a [[group of prime order]] <math>p</math> where <math>p</math> is odd, then <math>L</math> is an extraspecial group of order <math>p^3</math>, and exponent <math>p</math>. In other words, <math>L</math> is [[prime-cube order group:U3p|the group of unitary upper triangular matrices of order three over the prime field]].
 ==References==