en>K9re11: removed Category:Group theory using HotCat as there are already more specific subcategories

2014-11-15T20:16:19Z

removed Category:Group theory using HotCat as there are already more specific subcategories

en>Zundark: revert - it meant what it said (and "characteristic group" isn't defined here anyway)

2014-02-15T15:27:38Z

revert - it meant what it said (and "characteristic group" isn't defined here anyway)

← Older revision		Revision as of 17:27, 15 February 2014
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	In [[mathematics]], particularly in the area of [[abstract algebra]] known as [[group theory]], a '''characteristic subgroup''' is a [[subgroup]] that is [[invariant (mathematics)\|invariant]] under all [[automorphism]]s of the parent [[group (mathematics)\|group]].<ref>{{cite book \| last1=Dummit \| first1=David S. \| last2=Foote \| first2=Richard M. \| title=Abstract Algebra \| publisher=[[John Wiley & Sons]] \| year=2004 \| edition=3rd \| isbn=0-471-43334-9}}</ref><ref>{{cite book \| last=Lang \| first=Serge \| authorlink=Serge Lang \| title=Algebra \| publisher=[[Springer Science+Business Media\|Springer]] \| series=[[Graduate Texts in Mathematics]] \| year=2002 \| isbn=0-387-95385-X}}</ref> Because [[inner automorphism\|conjugation]] is an automorphism, every characteristic subgroup is [[normal subgroup\|normal]], though not every normal subgroup is characteristic. Examples of characteristic subgroups include the [[commutator subgroup]] and the [[center of a group]].		Nice to meet you, I am Marvella Shryock. He utilized to be unemployed but now he is a meter reader. Doing ceramics is what my family and I enjoy. North Dakota is her beginning place but she will have to transfer 1 day or an additional.<br><br>Here is my site: [http://Www.Rll.cc/dietdelivery10007 http://Www.Rll.cc]

	~~== Definitions ==~~
	~~A '''characteristic subgroup''' of a [[group (mathematics)\|group]] ''G'' is a [[subgroup]] ''H'' that is invariant under each [[automorphism]] of ''G''. That is,~~
	~~:<math>\varphi(H) = H</math>~~
	~~for every automorphism ''φ'' of ''G'' (where ''φ''(''H'') denotes the [[Image (mathematics)\|image]] of ''H'' under ''φ'').~~

	~~The statement “''H'' is a characteristic subgroup of ''G''” is written~~
	~~:<math>H\;\mathrm{char}\;G.</math>~~

	~~== Characteristic vs. normal ==~~
	~~If ''G'' is a group, and ''g'' is a fixed element of ''G'', then the conjugation map~~
	~~:<math>x \mapsto g x g^{-1}</math>~~
	is an automorphism of ''G'' (known as an [[inner automorphism]]). A subgroup of ''G'' that is invariant under all inner automorphisms is called [[normal subgroup\|normal]]. Since a characteristic subgroup is invariant under all automorphisms, every characteristic subgroup is normal.

	~~Not every normal subgroup is characteristic. Here are several examples:~~
	* Let ''H'' be a group, and let ''G'' be the [[direct product of groups\|direct product]] ''H'' × ''H''. Then the subgroups {1} × ''H'' and ''H'' × {1} are both normal, but neither is characteristic. In particular, neither of these subgroups is invariant under the automorphism (''x'', ''y'') → (''y'', ''x'') that switches the two factors.
	* For a concrete example of this, let ''V'' be the [[Klein four-group]] (which is [[group isomorphism\|isomorphic]] to the direct product [[cyclic group\|'''Z'''<sub>2</sub>]] × [[cyclic group\|'''Z'''<sub>2</sub>]]). Since this group is [[abelian group\|abelian]], every subgroup is normal; but every permutation of the three non-identity elements is an automorphism of ''V'', so the three subgroups of order 2 are not characteristic.Here <math> V=\left\{e,a,b,ab\right\}</math> Consider H={e,a} and consider the automorphism <math> T(e)=e, T(a)=b, T(b)=a, T(ab)=ab </math>.Then ''T(H)'' is not contained in ''H''.
	* In the [[quaternion group]] of order 8, each of the cyclic subgroups of order 4 is normal, but none of these are characteristic. However, the subgroup {1, −1} is characteristic, since it is the only subgroup of order 2.
	~~Note: If ''H'' is the unique subgroup of a group ''G'', then ''H'' is characteristic in ''G''.~~
	* If ''n'' is even, the [[dihedral group]] of order 2''n'' has three subgroups of [[index of a subgroup\|index]] two, all of which are normal. One of these is the cyclic subgroup, which is characteristic. The other two subgroups are dihedral; these are permuted by an [[outer automorphism group\|outer automorphism]] of the parent group, and are therefore not characteristic.
	* "Normality" is not transitive, but Characteristic has a transitive property, ~~namely if ''H'' Char ''K'' and ''K'' normal in ''G'' then ''H'' normal in ''G''~~.

	~~== Comparison~~ to ~~other subgroup properties ==~~
	~~=== Distinguished subgroups ===~~
	A related concept is that of a '''distinguished subgroup'''. In this case the subgroup ''H'' is invariant under the applications of [[surjective]] [[endomorphism]]s. For a [[finite group]] this is the same, because surjectivity implies injectivity, but ~~not for an infinite group: a surjective endomorphism is not necessarily an automorphism.~~

	~~=== Fully invariant subgroups ===~~
	For an even stronger constraint, a [[fully characteristic subgroup]] (also called a '''fully invariant subgroup''') ''H'' of a group ''G'' is a group remaining invariant under every endomorphism of ''G''; in other words, if ''f'' : ''G'' → ''G'' is any [[group homomorphism\|homomorphism]], then ''f''(''H'') is a subgroup of ''H''.

	~~=== Verbal subgroups ===~~
	~~An even stronger constraint is [[verbal subgroup]], which~~ is ~~the image of a fully invariant subgroup of a [[free group]] under~~ a ~~homomorphism~~.

	~~=== Containments ===~~
	~~Every subgroup that is fully characteristic~~ is ~~certainly distinguished~~ and ~~therefore characteristic; but a characteristic or even distinguished subgroup need not be fully characteristic~~.

	~~The [[center of a group]]~~ is ~~always a distinguished subgroup,~~ but ~~it is not always fully characteristic. The finite group of order 12, Sym(3) × '''Z'''/2'''Z''' has a homomorphism taking (''π'', ''y'')~~ to ~~( (~~1~~,2)<sup>''y''</sup>, ''0'') which takes the center 1 × '''Z'''/2'''Z''' into a subgroup of Sym(3) × 1, which meets the center only in the identity~~.

	~~The relationship amongst these subgroup properties can be expressed as:~~

	~~:[[subgroup]] ⇐ [[normal subgroup]] ⇐ '''characteristic subgroup''' ⇐ distinguished subgroup ⇐ [[fully characteristic subgroup]] ⇐ [[verbal subgroup]]~~

	~~==Examples==~~
	~~=== Finite example ===~~
	~~Consider the group ''G'' = S~~<~~sub~~>3<~~/sub~~> ~~× Z<sub>2</sub> (the group of order 12 which~~ is ~~the direct product of the [[symmetric group]] of order 6 and a [~~[cyclic group]] of order 2). The center of ''G'' is its second factor Z<sub>2</sub>. Note that the first factor S<sub>3</sub> contains subgroups isomorphic to Z<sub>2</sub>, for instance {identity,(12)}; let ''f'': ~~Z<sub>2</sub> → S<sub>3<~~/~~sub> be the morphism mapping Z<sub>2<~~/sub> onto the indicated subgroup. Then the composition of the projection of ''G'' onto its second factor Z<sub>2</sub>, followed by ''f'', followed by the inclusion of S<sub>3</sub> into ''G'' as its first factor, provides an endomorphism of ''G'' under which the image of the center Z<sub>2</sub> is not contained in the center, so here the center is not a fully characteristic subgroup of ''G''.

	~~=== Cyclic groups ===~~
	~~Every subgroup of a cyclic group is characteristic.~~

	~~=== Subgroup functors ===~~
	~~The [[derived subgroup]] (or commutator subgroup) of a group is a verbal subgroup. The [[torsion subgroup]] of an [[abelian group]] is a fully invariant subgroup.~~

	~~=== Topological groups ===~~
	~~The [[identity component]] of a [[topological group]] is always a characteristic subgroup~~.

	~~== Transitivity ==~~
	The property of being characteristic or fully characteristic is [[transitive relation\|transitive]]; if ''H'' is a (fully) characteristic subgroup of ''K'', and ''K'' is a (fully) characteristic subgroup of ''G'', then ''H'' is a (fully) characteristic subgroup of ''G''.

	Moreover, while it is not true that every normal subgroup of a normal subgroup is normal, it is true that every characteristic subgroup of a normal subgroup is normal. Similarly, while it is not true that every distinguished subgroup of a distinguished subgroup is distinguished, it is true that every fully characteristic subgroup of a distinguished subgroup is distinguished.

	~~==Map on Aut and End==~~
	~~If <math>H\,\mathrm{char}\,G.<~~/~~math>, then every automorphism of ''G'' induces an automorphism of the quotient group ''G~~/~~H'', which yields a map <math>\mbox{Aut}\,G \to \mbox{Aut}\, G/H<~~/~~math>~~.

	~~If ''H'' is fully characteristic in ''G'', then analogously, every endomorphism of ''G'' induces an endomorphism of ''G/H'', which yields a map~~
	~~<math>\mbox{End}\,G \to \mbox{End}\, G/H</math>~~.

	~~==See also==~~
	* [[Characteristically simple group]]

	~~==References==~~
	~~{{reflist}}~~

	~~[[Category:Group theory]]~~
	~~[[Category:Subgroup properties]~~]

173.25.54.191: del 'easily seen to be'

2013-03-09T02:30:00Z

del 'easily seen to be'

Show changes

157.181.226.93: example -- identity component

2012-06-15T07:46:34Z

example -- identity component

Show changes

Characteristic subgroup - Revision history

en>K9re11: removed Category:Group theory using HotCat as there are already more specific subcategories

en>Zundark: revert - it meant what it said (and "characteristic group" isn't defined here anyway)

173.25.54.191: del 'easily seen to be'

157.181.226.93: example -- identity component