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Topic: Automorphism

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	Automorphism - Wikipedia, the free encyclopedia
		In graph theory an automorphism of a graph is a permutation of the nodes that preserves edges and non-edges.
		An automorphism of a differentiable manifold M is a diffeomorphism from M to itself.
		In Riemannian geometry an automorphism is a self-isometry.
	en.wikipedia.org /wiki/Automorphism (898 words)

	PlanetMath: automorphism
		Roughly, an automorphism is a map from a mathematical object onto itself such that: 1.
		In the category of topological spaces an automorphism would be a bijective, continuous map such that its inverse map is also continuous (not guaranteed as in the group case).
		This is version 5 of automorphism, born on 2003-07-28, modified 2005-04-14.
	www.planetmath.org /encyclopedia/Automorphism4.html (196 words)

	Automorphism (Site not responding. Last check: 2007-10-07)
		Very informally, an automorphism is a symmetry of the object, a way of showing its internal regularity (whichever side of a regular polygon you choose as it basis, it looks the same).
		For example, in graph theory an automorphism of a graph is a permutation of the nodes that maps the graph to itself.
		The set of automorphisms of an object X together with the operation of function composition forms a group called the automorphism group of X, Aut(X).
	www.bopedia.com /en/wikipedia/a/au/automorphism_1.html (311 words)

	Outer automorphism group - Wikipedia, the free encyclopedia
		In mathematics, the outer automorphism group of a group G is the quotient of the automorphism group Aut(G) by its inner automorphism group Inn(G).
		The outer automorphism group of a finite simple group of Lie type is an extension of a group of "diagonal automorphisms" (cyclic except for D
		The outer automorphism group of a finite simple group in some infinite family of finite simple groups can almost always be given by a uniform formula that works for all elements of the family.
	en.wikipedia.org /wiki/Outer_automorphism_group (1004 words)

	Inner automorphism - Wikipedia, the free encyclopedia
		By associating the element a in G with the inner automorphism f in Inn(G) as above, one obtains an isomorphism between the factor group G/Z(G) (where Z(G) is the center of G) and Inn(G).
		At the opposite end of the spectrum, it is possible that the inner automorphisms exhaust the entire automorphism group---a group whose automorphisms are all inner is called complete.
		The notion of inner automorphism for Lie algebras is compatible with the notion for groups in the sense that an inner automorphism of a Lie group induces a unique inner automorphism of the corresponding Lie algebra.
	en.wikipedia.org /wiki/Inner_automorphism (312 words)

	PlanetMath: inner automorphism
		It is easy to show the conjugation map is in fact, a group automorphism.
		An automorphism that isn't inner is called an outer automorphism.
		This is version 7 of inner automorphism, born on 2002-07-04, modified 2003-02-25.
	planetmath.org /encyclopedia/Inner.html (99 words)

	Creation of Automorphism Groups
		An automorphism group of the finite group G may be created in one of two ways.
		Secondly, an arbitrary group of automorphisms A of G may be created by giving a set of generators for A defined in terms of their action on a set of generators for G. AutomorphismGroup(G): Grp -> GrpAuto
		When G is a non-soluble permutation or matrix group, the algorithm relies on a database of automorphism groups for the non-cyclic simple factors of G, hence the non-abelian composition factors of G must belong to a restricted list.
	www.math.lsu.edu /magma/text364.htm (453 words)

	Automorphism (Site not responding. Last check: 2007-10-07)
		Very informally, an automorphism is a symmetry of the object, a way of showingits internal regularity (whichever side of a regular polygon you choose as it basis,it looks the same).
		For example, in graph theory an automorphism of a graph is a permutationof the nodes that maps the graph to itself.
		In group theory, an automorphism of a group G is a bijective homomorphism of G ontoitself (that is, a one-to-one map G
	www.therfcc.org /automorphism-32663.html (313 words)

	PlanetMath: outer automorphism group
		The outer automorphism group of a group is the quotient of its automorphism group by its inner automorphism group:
		"outer automorphism group" is owned by Thomas Heye.
		This is version 8 of outer automorphism group, born on 2003-10-15, modified 2004-03-11.
	www.planetmath.org /encyclopedia/OuterAutomorphismGroupOfAGroup.html (73 words)

	[No title]
		The main result states that the group of differentiable automorphisms of a differentiable double loop is compact with respect to the compact-open topology.
		It is proved that the automorphism group of a locally compact connected double loop is a locally compact transformation group with respect to the compact-open topology.
		In particular, every continuous automorphism of a smooth polygon is smooth, and thus the topological automorphism group of the polygon, endowed with the compact-open topology, is a smooth Lie transformation group.
	www.mathematik.uni-tuebingen.de /ab/Geometrie.alt/Abstracts.html (1815 words)

	The Automorphism Group of an Incidence Structure
		The automorphism group A of an incidence structure D is always presented as a permutation group G acting on the standard support.
		A cyclic subgroup H of the automorphism group G of the incidence structure D. The purpose of this function is to terminate the search for automorphisms of D as soon as a non--trivial automorphism is found.
		As noted at the beginning of the section, the automorphism group G of an incidence structure D is given in its action on the standard support and it does not act directly on D. The action of G on D is obtained using the G--set mechanism.
	www.math.niu.edu /help/math/magmahelp/text1149.html (1223 words)

	Automorphism Group of a Graph or Digraph
		The automorphism group functionality is an implementation of B. McKay's nauty programme.
		The automorphism group A of a graph G is given in its action on the standard support and it does not act directly on G. The action of A on G is obtained using the G--set mechanism.
		Let a be an element of the automorphism group A for the graph G and let Y be a G--set for A. Given an element y belonging either to Y or to a G--set derived from Y, find the image of y under a.
	www.math.niu.edu /help/math/magmahelp/text1132.html (2409 words)

	Automorphisms (Site not responding. Last check: 2007-10-07)
		Given a map object m from G to G, which is an isomorphism, returns the associated automorphism as an automorphism of a group of Lie type.
		The diagonal automorphism of the semisimple group of Lie type G given by the vector v.
		The function returns the automorphism given by the character chi defined by chi(alpha_i)=v_i, where alpha_i is the ith simple root.
	www.math.lsu.edu /magma/text1053.htm (415 words)

	Inner Automorphisms (Site not responding. Last check: 2007-10-07)
		An automorphism that is not an inner automorphism is an outer automorphism.
		The composition of two inner automorphisms, derived from x and y, is the inner automorphism derived from yx.
		Thus the inner automorphisms form a subgroup of all the automorphisms of g.
	www.mathreference.com /grp,inner.html (388 words)

	M. Pettet publications (Site not responding. Last check: 2007-10-07)
		On the automorphism tower of a Cernikov group, J. Lond.
		Groups whose automorphisms are almost determined by their restriction to a subgroup, Glasgow Math.
		Central automorphisms of periodic groups, Archiv der Math., 51 (1988), 10-33; MR 90a:20057.
	www.math.utoledo.edu /~mpettet/publications.html (346 words)

	Physics Help and Math Help - Physics Forums - how to find automorphism?
		An automorphism of a group G is a bijective homomorphism from the group G to itself.
		I'd have thought f(x)= -x is a automorphism from Z to Z. But given what has been said here I have tried to figure out why it isn't one, but with no sucess....
		Then the map G to G, given by f(y)=xyx^{-1} is an automorphism that is not the identity automorphism (or x would commute with all elements #).
	www.physicsforums.com /printthread.php?t=53201 (767 words)

	[No title]
		The set of all automorphisms of a given graph is the automorphism group of the graph.
		In particular, the orbits of an automorphism group identify symmetrical vertices.
		However, in the context of chemistry because molecules are a restricted class of graphs, we have proven the problems of graph isomorphism, automorphism partitioning, and canonical labeling to be polynomial-time.
	www.cs.sandia.gov /~jfaulon/MICS/isomorphism/isomorphism.html (610 words)

	ABSTRACTS DROSTE
		We describe the normal subgroup lattice of the automorphism groups of the countable universal homogeneous distributive lattice and of the countable atomless generalized Boolean lattice.
		For both finite and infinite chain cases the simple automorphism groups split into two classes: those where there is a bound (<12) on the number of conjugates required to express one non-identity element in terms of another, and those in which there is no such bound.
		It is shown that the automorphism group has a smallest non-trivial normal subgroup, a largest proper normal subgroup, and at least $2^2^\omega$ normal subgroups between these two.
	www.informatik.uni-leipzig.de /~droste/droabal.html (3181 words)

	FREE GROUPS
		These are problems about free groups, their automorphisms and related issues.
		Lubotzky and A. Lue have proved that every normal automorphism of a free group is inner.
		(F32) An automorphism of a free group F is called an IA-automorphism if it is Identical on the Abelianization F/[F, F].
	zebra.sci.ccny.cuny.edu /web/nygtc/problems/probfree.html (1478 words)

	That Logic Blog: Semantics is (coNP) hard, syntax is (GI) easy
		The group of all automorphisms of f is denoted by Aut(f).
		A syntactic automorphism is a permutation of the literals appearing in p, which preserves the syntactic structure (so, swapping two clauses counts as an automorphism).
		I mentioned in the first post that it is possible to construct a coloured graph whose automorphism group is the same as the syntactic automorphism group of p.
	thatlogicblog.blogspot.com /2005/03/semantics-is-conp-hard-syntax-is-gi.html (718 words)

	3-D Crystals XVII
		An automorphism is an isomorphism of a group with itself.
		In fact an automorphism is a permutation of the group elements such that the structure (of the table) is preserved.
		Geometrically this (inner) automorphism can be interpreted as an interchange of the mirrors a and c together with an interchange of the mirrors b and d.
	home.hetnet.nl /~turing/d3_lattice_17.html (2565 words)

	The Peace Encyclopedia: Automorphism, Ethnomorphism
		Thus 'automorphism' is the psychology of projecting one's own qualities, emotions, values or culture onto other people.
		And 'ethnomorphism' is the psychology of projecting the characteristics and values of one's own ethnic group onto other ethnic groups.
		More dangerously, automorphism can result in the inability to comprehend and acknowledge in others what is commonly known as 'evil' except, perhaps, in hindsight.
	www.yahoodi.com /peace/automorphism.html (2044 words)

	SIGGS Theory Model: Automorphism (Site not responding. Last check: 2007-10-07)
		Automorphism in a system allows components and connections to be rearranged, while continuing to allow the system to functionas before.
		Automorphism is more easily comprehended when considering team or group settings, in which individuals perform several tasks.
		If educational system automorphism increases, then input increases and storeput increases and fromput increases and feedout decreases and filtration decreases and spillage decreases and efficiency decreases.
	www.indiana.edu /~educr547/frick/automorp.html (213 words)

	Fraunhofer ITWM: Automorphism Groups of Hyperelliptic Function Fields
		In the project "Hyperelliptic Curve Cryptography" researchers at Fraunhofer ITWM developed an efficient method to compute the automorphism group* of an arbitrary hyperelliptic function field*.
		The Jacobians* of hyperelliptic function fields* have been suggested by Nean Koblitz in 1988 as groups* for cryptographic purposes, because the computation of the discrete logarithm* is believed to be hard in this kind of groups.
		A large number of examples of automorphism groups of hyperelliptic function fields is available here (ZIP-file, 601k, uncompressed size: 5.9M).
	www.itwm.fhg.de /mab/competences/Crypto/aut?language=en (306 words)

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