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Topic: Characteristic subgroup

	PlanetMath: characteristic subgroup
		This is precisely the definition of a normal subgroup.
		This is version 10 of characteristic subgroup, born on 2002-07-21, modified 2006-02-17.
		For the phrase, "H is a characteristic subgroup of G", the only notation I have ever seen in books and used by professors is "H char G".
	planetmath.org /encyclopedia/Characteristic3.html (187 words)

	Normal Structure and Characteristic Subgroups (Site not responding. Last check: 2007-10-30)
		Given a finite p-group G, return the characteristic subgroup of G generated by the elements x^(p^i), x in G, where i is a positive integer.
		Given a finite p-group G, return the characteristic subgroup of G generated by the elements of order dividing p^i, where i is a positive integer.
		Given a group G and a subgroup H of G, return a sequence of subgroups commencing with G and terminating with H, such that each subgroup is normal in the previous one.
	www.math.uiuc.edu /Software/magma/text200.html (293 words)

	Characteristic subgroup - Groupprops
		That is, the intersection of a family of characteristic subgroups is characteristic.
		Since characteristicity is an invariance property with respect to a property stronger than that of being an endomorphism, the property of being characteristic is join-closed.
		The subgroup property of being characteristic is a quotient-transitive subgroup property.
	groupprops.wiki-site.com /index.php/Characteristic_subgroup (910 words)

	Characteristic subgroup - Definition, explanation
		In abstract algebra, a characteristic subgroup of a group G is a subgroup H of G invariant under each automorphism of G.
		The derived subgroup (or commutator subgroup) of a group is always a fully characteristic subgroup, as is the torsion subgroup of an abelian group.
		The property of being characteristic or fully characteristic is 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.
	www.calsky.com /lexikon/en/txt/c/ch/characteristic_subgroup.php (420 words)

	Fully characteristic subgroup - Groupprops
		A subgroup of a group is termed fully characteristic or fully invariant if it is invariant under all endomorphisms of the whole group.
		That is, the intersection of a family of fully characteristic subgroups is fully characteristic.
		The property of being fully characteristic is join-closed, viz an arbitrary join (subgroup generated) of fully characteristic subgroups is fully characteristic.
	groupprops.wiki-site.com /index.php/Fully_characteristic_subgroup (472 words)

	Characteristic subgroup - Wikipedia, the free encyclopedia
		In mathematics, a characteristic subgroup of a group G is a subgroup H that is invariant under each automorphism of G.
		The derived subgroup (or commutator subgroup) of a group is always a fully characteristic subgroup, as is the torsion subgroup of an abelian group.
		The property of being characteristic or fully characteristic is 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.
	en.wikipedia.org /wiki/Characteristic_subgroup (565 words)

	PlanetMath: fully invariant subgroup
		This is a stronger condition than being a characteristic subgroup.
		See Also: characteristic subgroup, subnormal subgroup, example of fully invariant subgroup
		This is version 3 of fully invariant subgroup, born on 2002-12-07, modified 2004-03-22.
	www.planetmath.org /encyclopedia/FullyCharacteristicSubgroup.html (68 words)

	Characteristic - Wikipedia, the free encyclopedia
		Characteristic is also sometimes used as a piece of jargon in discussions of universals in metaphysics, often in the phrase 'distinguishing characteristics'.
		An I-V or current-voltage characteristic is the current in a circuit as a function of the applied voltage
		The characteristic is also the distinguishing flash pattern of a lighthouse meant to be seen at night.
	en.wikipedia.org /wiki/Characteristic (120 words)

	Springer Online Reference Works
		The set of fully-characteristic subgroups forms a sublattice in the lattice of all subgroups.
		The commutator subgroup and the members of the lower central series in an arbitrary group are fully-characteristic subgroups.
		In addition, any verbal subgroup of a group is fully characteristic.
	eom.springer.de /F/f041920.htm (113 words)

	[No title]
		By the depth of a characteristic subgroup H of G, we will mean the largest integer d such that H lies in p^d G. The preceding lemmas tell us about characteristic subgroups of depth 0.
		Every characteristic subgroup H of G is a sum of characteristic subgroups of G of the form G(i,j).
		Therefore p^e H is the sum of subgroups of p^e G of the form (p^e G)(m,n).
	www.math.niu.edu /~rusin/known-math/99/characteristic (1815 words)

	[No title]
		A finite nilpotent group in which the lattice of normal subgroups is a chain is a cyclic $p$-group.
		Then the normal subgroups of G are its p + 1 maximal subgroups and the terms of the lower central series.
		Subject: Characteristic maximal subgroups in groups of maximal class Avinoam Mann has noted that in the "exceptional" $p$-groups of maximal class there are two distinct characteristic maximal subgroups, so such groups do not have CSC (Characteristic Subgroups form a Chain).
	www.bath.ac.uk /~masgcs/problem/commentary10.html (2226 words)

	Normal Structure and Characteristic Subgroups
		The maximal subgroups of the finite group G returned as a sequence of subgroups.
		The subgroups of the finite group G returned as a sequence of records.
		is set, only subgroups such that the quotient group has invariants equal to the given sequence are found.
	www.umich.edu /~gpcc/scs/magma/text285.htm (551 words)

	Brussels Griffon - Breed overview. substance, griffin, terrier, toy, cheeks, pets, coach, characteristic, netherlands, ... (Site not responding. Last check: 2007-10-30)
		Characteristic is also sometimes used as a piece of jargon in discussions of Universal--metaphysics in metaphysics, often in the phrase 'distinguishing characteristics '.
		The letter seemed no more than a characteristic outburst from goethe park impulsive girl who had but yesterday fallen asleep at evening in his arms.
		It is thought that the dense population of an endemic breed of large and fierce dogs was the characteristic that most struck the few ancient romans who established contact with the islands by the sea.
	www.grifon.info /Substance.html (1000 words)

	Normal Structure and Characteristic Subgroups
		Given a p-group G, return the characteristic subgroup of G generated by the elements x^(p^i), x in G, where i is a positive integer.
		A composition series for the group G. The series is returned as a sequence of subgroups of G. The i-th term of the composition series has a presentation given by the generators G.i through G.NPCgens(G) and relations involving those generators only.
		Given a p-group G, return the characteristic subgroup of G generated by the elements of order dividing p^i, where i is a positive integer.
	www.math.ufl.edu /help/magma/text247.html (557 words)

	Vipul Naik's Home Page: Unsolved Problems
		Normal and potentially characteristic subgroups: Call a subgroup of a group potentially characteristic if there is a group containing the bigger group in which the subgroup is characteristic.
		Intersection of central factors A central factor of a group is a subgroup such that every inner automorphism of the group restricts to an inner automorphism of the subgroup.
		A subgroup of a group is said to be strictly characteristic or distinguished if every surjective endomorphism of the group restricts to an endomorphism of the subgroup.
	www.cmi.ac.in /~vipul/unsolvedproblems (609 words)

	Analog Surround Mixing Console SRC51 and 5.1 Systems
		Audio recording while using the same channel for monitoring the output of a track is possible with or without the internal processing devices.
		If it is used as an audio subgroup, the buss amp can be assigned to the input of the unit or to the input of the monitor chain.
		With this combination it is possible to use the entire module as a groupmaster, or to use only the monitor chain as a group master and the rest of the channel either for recording in direct mode or to feed the 'Tape' or 'Line' input to a mono or stereo audio subgroup.
	www.surroundconsole.com (751 words)

	[No title]
		The PJ are the parabolic subgroups of G(q).
		Hence by Lemma 6.1, the subgroups of L \ CU(oe) 2 Syl2(L) which are radical in * L are exactly the subgroups* L \ CUJ(oe) for ø-invariant subsets J I. Since I has ju* st two ø-orbits, this means that there are exactly two such proper subgroups*.
		Thus for any elementary abelian 2-subgroup E L, there is a subgroup E0 E such that all involutions in E0 are of type 2B in G and all involutions in Er E* *0 are of type 2A, and such that rk(E0) 2 and rk(E=E0) 3.
	www.math.purdue.edu /research/atopology/Oliver/limz.txt (21296 words)

	Ordlista
		Repeatability means the variation when an operator uses the same instrument to measure the same characteristic several times.
		Rbar R, the mean (average) of all subgroup ranges.
		Xbar X, the mean (average) of all values or of the values in a subgroup.
	en.rektron.se /dictionary.aspx (820 words)

	Passman's Abstracts
		Specifically, we show that certain basic results which hold when G is a polycyclic-by-finite group with no finite normal subgroups need not hold in the case of group algebras of finite groups.
		Here we begin the analysis in the case where the abelian group A is the additive group of a finite-dimensional vector space V over a locally finite field F of prime characteristic p, and the automorphism group G is a simple infinite absolutely irreducible subgroup of GL(V).
		We also consider simple Artinian rings with involution, in characteristic not 2, and we determine those bounded Z-filtrations that are maximal subject to being stable under the action of the involution.
	www.math.wisc.edu /~passman/abstracts.html (3706 words)

	Math Forum Discussions
		Suppose G is any group with K characteristic in G. Suppose H < K is any
		subgroup, what can be said about H with relation to G? Is H characteristic in G? is it normal in G? Thank you.
		The Math Forum is a research and educational enterprise of the Drexel School of Education.
	www.mathforum.org /kb/thread.jspa?threadID=1448307&messageID=5123557 (87 words)

	Math 720
		The only subgroups of G are G itself and the identity.
		Let G be a group, we define the commutator subgroup of G to be the subgroup of G generated by elements of the form x
		(Such a subgroup is called a characteristic subgroup).
	www.ndsu.nodak.edu /ndsu/coykenda/M720.2.F2000.htm (187 words)

	[No title]
		The characteristic subgroup 1(P) of P is the subgroup generated by all order p elements.
		Local subgroup structure Throughout this section G is a finite group with extraspecial Sylow p-subgro* *up P.
		Recall that Op(G) is the subgroup generated by all p0-elements of G. Lemma 3.2.
	www.math.purdue.edu /research/atopology/GreenDJ-Minh/ogawa.txt (2428 words)

	Math Forum Discussions
		Let G be a finite group with a normal subgroup N. Let Phi(G) be the Frattini subgroup of G, defined as the intersection
		of all maximal proper subgroups of G - this is easily seen to be a
		Show that every Sylow subgroup of N is normal in G. Any hints comments or solutions are most welcome.
	www.mathforum.org /kb/message.jspa?messageID=53816&tstart=0 (130 words)

	Atlas: The structure of abelian Lie pro-Lie groups by Sidney A. Morris (Site not responding. Last check: 2007-10-30)
		A prodiscrete group is a complete abelian topological group in which the open normal subgroups form a basis of the filter of identity neighborhoods.
		It is likely that a group like this factor group is always a finitely generated free discrete group, as is proved to be the case if it is assumed to be first countable.
		The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cakb-24.
	atlas-conferences.com /cgi-bin/abstract/cakb-24 (231 words)

	S.O.S. Mathematics CyberBoard :: View topic - Derivative
		in fact) of G, every fully characteristic subgroup is normal.)
		is fully characteristic subgroup of G. Proof: let f be any endomorphism of G and
		characteristic subgroup is normal and clearly 2) holds for them.
	www.sosmath.com /CBB/viewtopic.php?t=27619 (241 words)

	[No title] (Site not responding. Last check: 2007-10-30)
		If $B$ is a characteristic subgroup of $W$, then we can view $Aut W$ as the normalizer in $Aut B$ of $W$.
		So we are interested in when $B$ is not characteristic.
		If $W$ is finite wtih $P\le S_n$ and $B$ is not characteristic, then $G= \langle u\rangle H$ with $H$ abelian of odd order, $u^2=1$, $h^u=h^{-1}$ for all $h\in H$, and $P$ contains a normal elementary abelian 2-group of size $2^{n/2}$; moreover the $Aut W$ orbit of $B$ has length 2 precisely.
	www.bath.ac.uk /~masgcs/problem/commentary28.html (168 words)

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