Show That an Intersection of Normal Subgroups of a Group $g$ Is Again a Normal Subgroup of $g$.

This commodity gives the statement, and possibly proof, of a subgroup property (i.e., normal subgroup) not satisfying a subgroup metaproperty (i.e., transitive subgroup holding).
View all subgroup metaproperty dissatisfactions | View all subgroup metaproperty satisfactions|Go help on looking upwards metaproperty (dis)satisfactions for subgroup backdrop
Get more than facts about normal subgroup|Get more facts about transitive subgroup property|

1 Statement
2 Fractional truth
- 2.1 Transitivity-forcing operator
- 2.ii Left transiter
- 2.iii Right transiter
- ii.4 Subnormality
3 Corollaries
4 Related facts
- 4.1 Making normality transitive
- four.2 For item kinds of groups
- iv.3 The extent of lack of transitivity
- 4.4 Analogues in other algebraic structures
- four.5 General tricks
5 Proof
- 5.1 Generic instance
- 5.2 Specific realizations of this generic example
six GAP implementation
- 6.ane Implementation of the generic example
- 6.two The implementation in some special cases
7 Listing of counterexamples of small gild
8 References
- eight.1 Textbook references
ix External links

Statement

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

Partial truth

Transitivity-forcing operator

Left transiter

While normality is not transitive, it is notwithstanding truthful that every feature subgroup of a normal subgroup is normal. Characteristicity is the left transiter of normality -- it is the weakest property $p$ such that every subgroup with property $p$ in a normal subgroup is normal. For full proof, refer: Feature 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. Beingness transitively normal is the right transiter of beingness normal. Backdrop like being a direct factor, being a central subgroup, and beingness a central factor imply being transitively normal.

Subnormality

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

A special case of this is the notion of two-subnormal subgroup, which is a normal subgroup of a normal subgroup. Another special case is the notion of a 3-subnormal subgroup, which is a normal subgroup of a normal subgroup of a normal subgroup.

There are also related notions of hypernormalized subgroup, 2-hypernormalized subgroup, ascendant subgroup, descendant subgroup, and series subgroup.

Corollaries

Normality is not a finite-relative-intersection-closed subgroup property, because finite-relative-intersection-airtight implies transitive.

Making normality transitive

For simplicity, nosotros assume $H \le K \le G$ , with $H$ the bottom grouping, $K$ the middle group, and $G$ the peak group.

Statement	Change in assumption	Change in determination
Characteristic of normal implies normal	$H$ a characteristic subgroup of $K$	$H$ is normal in $G$
Left transiter of normal is feature	$H$ in $K$ is such that for whatsoever possible $G$ with $K$ normal in $G$ , $H$ is normal in $G$	$H$ is feature in $K$
Equivalence of definitions of transitively normal subgroup	$H$ is normal in $K$ ; spelling out weather for $K$ in $G$ such that ...	$H$ is normal in $G$
Central gene implies transitively normal	$K$ a key factor of $G$	$H$ is normal in $G$
Direct cistron implies transitively normal	$K$ a direct gene of $G$	$H$ is normal in $G$

For particular kinds of groups

For simplicity, we refer beneath to the three groups as $H \le K \le G$ , with $H$ the lesser grouping, $K$ the center group, and $G$ the peak group, such that $H$ is normal in $K$ and $K$ is normal in $G$ , but $H$ is non normal in $G$ .

Stronger formulation	Boosted restrictions introduced	Boosted comments/examples
Normality is not transitive for any nontrivially satisfied extension-airtight group property	$H,K,G$ all satisfy a group property $\alpha$ closed under taking extensions	$\alpha$ could be the holding of being a finite $p$ -group, any fixed prime $p$ ; or solvability, or finiteness
Conjunction of normality with any nontrivial finite-direct product-airtight property of groups is non transitive	$H$ and $K$ both satisfy a group holding $\alpha$ that is closed under finite straight products	Abelian normal subgroup of abelian normal subgroup demand non exist normal
Every nontrivial normal subgroup is potentially 2-subnormal-and-not-normal	We are given $H \le L$ and must detect $K,G$ with $G$ containing $L$
Normality is not transitive for any pair of nontrivial quotient groups	We are given nontrivial groups $A,B$ and must ensure $K/H \cong A, G/K \cong B$

The extent of lack of transitivity

Stronger formulation	Meaning of formulation	How "normality is non transitive" is a special case
There exist subgroups of arbitrarily large subnormal depth	For whatever natural number $n$ , there exists a group $G$ , subgroup $H$ , such that the shortest subnormal series for $H$ in $G$ has length $n$ . In other words, the minimum length of a concatenation from $H$ upwards to $G$ , with each subgroup normal in the next, is $n$	Case $n = 2$
Descendant non implies subnormal	There exist subgroups for which in that location is a descending chain from whole group to subgroup, each normal in predecessor, of countable length (so intersection of all members is subgroup) but no finite concatenation
at that place exist subgroups of arbitrarily large descendant depth
Ascendant not implies subnormal	There exist subgroups for which there is an ascending chain from subgroup to whole grouping, each normal in successor, of countable length (and so spousal relationship of all members is whole group) merely no finite chain
there exist subgroups of arbitrarily large ascendant depth
Normal non implies left-transitively fixed-depth subnormal	We can accept a normal subgroup $H$ of $K$ such that for every $k$ , there exists a grouping $G$ containing $K$ as $k$ -subnormal just $H$ is not $k$ -subnormal	Case $k = 1$
Normal not implies right-transitively fixed-depth subnormal	We can have a normal subgroup $K$ of $G$ such that for every $k$ , in that location exists a group $H$ that is $k$ -subnormal in $K$ , not in $G$	Instance $k = 1$

Analogues in other algebraic structures

Statement	Analogy correspondences	Additional comments
Ideal property is not transitive for Lie rings	Lie ring $\leftrightarrow$ group, ideal of a Prevarication ring $\leftrightarrow$ normal subgroup
Normality is not transitive for field extensions	field extension $\leftrightarrow$ group (namely, its Galois grouping), normal field extension $\leftrightarrow$ normal subgroup in Galois correspondence	A normal field extension of a normal field extension need not be normal. In fact, by the key theorem of Galois theory, this corresponds directly to the fact that a normal subgroup of a normal subgroup need not be normal.
Normality is not composition-airtight	normal monomorphism $\leftrightarrow$ normal subgroup	A composite of normal monomorphisms need not be normal.

General tricks

Disproving transitivity
Using dihedral groups as counterexamples

Proof

(Also see #List of counterexamples of pocket-size order).

Generic instance

A natural case is as follows. Take whatever nontrivial group $H$ , and consider the square, $K = H \times H$ (the external directly product of $H$ with itself). Now, consider the external semidirect product of this group with the grouping $\mathbb{Z}/2\mathbb{Z}$ (the circadian group of 2 elements) acting via the exchange automorphism (the automorphism that exchanges the coordinates). Call the big grouping $G$ .

(Annotation that $G$ can be described more than compactly every bit the external wreath production of $H$ with the group of order two interim regularly.)

Allow $H_1, H_2$ exist the copies of $H$ embedded in $K$ equally $H \times \{ e \}$ and $\{ e \} \times H$ . We then have:

Annotation that both $H_1$ and $H_2$ are copies of $H$ , and hence either can exist viewed as the Base of a wreath product (?) in $G$ .

SIDENOTE: This example is not, in some sense, an extreme case of normality not being transitive. In fact, the property of being the base of operations of a wreath product is transitive, and whatsoever base of a wreath product is a two-subnormal subgroup, which implies that applying this structure iteratively does not yield subgroups of subnormal depth more than ii. Even further, base of a wreath product implies right-transitively 2-subnormal, or equivalently, any two-subnormal subgroup of the base of a wreath product is ii-subnormal in the whole group.

Specific realizations of this generic case

The smallest instance of this yields $H_1$ a group of gild 2, and $G$ a group of order eight. In fact, hither $G$ is the dihedral group of club eight and $H$ is a cyclic grouping of order 2, with $H_1$ and $H_2$ being subgroups of guild two generated by reflections. Here'south how this relates to the usual definition of the dihedral grouping:

$G = \langle a,x \mid a^4 = x^2 = e, xax = a^{-1} \rangle$

$H_1 = \langle x \rangle, \qquad H_2 = \langle a^2 x \rangle, \qquad K = \langle x, a^2 \rangle$ .

Here, $H_1$ and $H_2$ are both normal in $K$ , which is normal in $G$ , but neither $H_1$ nor $H_2$ is normal in $G$ .

For more on the subgroup structure of the dihedral group, refer subgroup structure of dihedral group:D8, Klein iv-subgroups of dihedral grouping:D8, and non-normal subgroups of dihedral group:D8.

GAP implementation

Implementation of the generic example

Here is an implementation of the generic example, with any nontrivial grouping $H$ . Notation that you lot need to ascertain $H$ for GAP before executing the commands in this example! Double semicolons have been used to suppress the output here, since the output depends on the selection of $H$ (you can utilise single semicolons instead to brandish all the outputs).

We get-go construct the groups $H_1, H_2, K, G$ using the wreath product control:

gap> K := WreathProduct(H,SymmetricGroup(2));; gap> H1 := Image(Embedding(Yard,one));; gap> H2 := Epitome(Embedding(Yard,2));; gap> Thousand := Group(Matrimony(H1,H2));;

Adjacent, we bank check that $H_1$ and $H_2$ are subgroups of $K$ and $K$ is a subgroup of $G$ :

gap> IsSubgroup(Yard,K); truthful gap> IsSubgroup(One thousand,H1); true gap> IsSubgroup(One thousand,H2); true

Finally, we bank check that $H_1, H_2$ are both normal in $K$ and $K$ is normal in $G$ , but $H_1$ and $H_2$ are not normal in $G$ .

gap> IsNormal(G,Chiliad); truthful gap> IsNormal(K,H1); truthful gap> IsNormal(Yard,H2); true gap> IsNormal(G,H1); simulated gap> IsNormal(G,H2); false

The implementation in some special cases

Hither is the implementation when $H$ is cyclic of order ii:

gap> Yard := WreathProduct(H,SymmetricGroup(2)); <group of size 8 with two generators> gap> H1 := Image(Embedding(G,1)); <group with 1 generators> gap> H2 := Image(Embedding(G,2)); <group with 1 generators> gap> K := Grouping(Wedlock(H1,H2)); <grouping with three generators> gap> IsSubgroup(G,Grand); true gap> IsSubgroup(K,H1); true gap> IsSubgroup(K,H2); true gap> IsNormal(Grand,K); true gap> IsNormal(Thousand,H1); truthful gap> IsNormal(K,H2); truthful gap> IsNormal(Grand,H1); fake gap> IsNormal(G,H2); false

Listing of counterexamples of small order

Big group	Order of large group	Violation of normality beingness transitive
dihedral grouping:D8	$8$	Klein 4-subgroup is normal, has normal subgroup of club two that is not normal in the whole grouping.
alternate group:A4	$12$	The normal Klein four-group comprising the identity and three double transpositions has a normal subgroup of order two that is not normal in the whole group.
SmallGroup(xvi,3)	$16$
SmallGroup(16,iv)	$16$
M16	$16$
dihedral group:D16	$16$
semidihedral group:SD16	$16$
quaternion grouping:Q16	$16$

References

Textbook references

Groups and representations past Jonathan Lazare Alperin and Rowen B. Bell, ISBN 0387945261, Folio 8, ^{More info} Also, Page 6 (offset mention), and Folio 17 (further explanation)
Abstract Algebra by David S. Dummit and Richard M. Foote, 10-digit ISBN 0471433349, 13-digit ISBN 978-0471433347, Folio 91, Department 3.2 (More on cosets and Lagrange's theorem), Example (3), (example of the dihedral group)^{More than info} As well, Page 135, with justification of the related fact that characteristic of normal implies normal
An Introduction to Abstract Algebra by Derek J. South. Robinson, ISBN 3110175444^{More than info}, Page 66
A Course in the Theory of Groups by Derek J. Due south. Robinson, ISBN 0387944613, Page 17, Exercise 1.iii.15, ^{More info} Too: Page 28, Page 63
Algebra by Michael Artin, ISBN 0130047635, xiii-digit ISBN 978-0130047632, Page 236, Miscellaneous Problems (Chapter six), Practice iv, (starred problem)^{More info}

External links

Search for "normality+is+non+transitive" on the World wide web:
Scholarly articles: Google Scholar, JSTOR
Books: Google Books, Amazon
This wiki: Internal search, Google site search
Encyclopaedias: Wikipedia (or using Google), Citizendium
Math resource pages:Mathworld, Planetmath, Springer Online Reference Works
Math wikis: Topospaces, Diffgeom, Commalg, Noncommutative
Discussion fora: Mathlinks, Google Groups
The web: Google, Yahoo, Windows Alive
Learn more than nearly using the Searchbox OR provide your feedback

perrywitelf.blogspot.com

Source: https://groupprops.subwiki.org/wiki/Normality_is_not_transitive