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Topic: Automorphism group

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	Automorphism - Wikipedia, the free encyclopedia
		A group automorphism is a group isomorphism from a group to itself.
		In Riemannian geometry an automorphism is a self-isometry.
		Conjugation by a is the group homomorphism φ
	en.wikipedia.org /wiki/Automorphism (898 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).
		This group is important in the topology of surfaces because there is a happy connection: the extended mapping class group of the surface is the Out of its fundamental group.
		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)

	Automorphism Group of a Graph or Digraph
		The automorphism group functionality is an implementation of B. McKay's nauty programme.
		Returns the automorphism group of the graph G in its action on the edges of G, and the G--set of edges of G. The parameters are almost identical to those for AutomorphismGroup (with the exception of
		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.
	www.math.niu.edu /help/math/magmahelp/text1132.html (2409 words)

	Representations of an Automorphism Group
		Construct a permutation representation of the group of automorphisms A. The function finds a union of conjugacy classes of the base group G which is closed under the action of A and with G-normal closure equal to G. The permutation action of A on such a set is faithful.
		Given a group of automorphisms A of a group G, this function returns the set of elements of G (i.e., a union of conjugacy classes) used as the support of the permutation group constructed by the
		A presentation for the group of automorphisms A on the generators of A. The isomorphism from the finitely presented group to the group of automorphisms A is also returned.
	www.math.lsu.edu /magma/text367.htm (419 words)

	Automorphisms (Site not responding. Last check: 2007-11-07)
		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 graph automorphism of the group of Lie type G given by the permutation p.
		The field automorphism of the group of Lie type G induced by sigma, an element of the automorphism group of the base field of G
	www.math.lsu.edu /magma/text1053.htm (415 words)

	PlanetMath: outer automorphism group (Site not responding. Last check: 2007-11-07)
		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.
	planetmath.org /encyclopedia/OuterAutomorphismGroupOfAGroup.html (73 words)

	Monster group (Site not responding. Last check: 2007-11-07)
		It is a simple group meaning it does not have any normal subgroups except for the subgroup consisting only the identity element and M itself.
		The finite simple groups have been completely there are several infinite families of finite groups plus 26 sporadic groups that don't follow any apparent pattern.
		The Monster was predicted by B. Fischer R. Griess in 1973 and first constructed Griess in 1980 as the automorphism group of the Griess algebra a 196883-dimensional nonassociative algebra.
	www.freeglossary.com /Monster_group (343 words)

	Group isomorphism (Site not responding. Last check: 2007-11-07)
		In abstract algebra, given two groups (G, ) and (H, @) a groupisomorphism from (G, ) to (H, @) is a bijective group homomorphism from G to H.
		The group Z of integers (with addition) is a subgroup of R, and the factor group R/Z is isomorphic to the group S
		The composition of twoautomorphism is again an automorphism, and with this operation the set of all automorphisms of a group G, denoted byAut(G), forms itself a group, the automorphism group of G.
	www.therfcc.org /group-isomorphism-190442.html (292 words)

	PlanetMath: inner automorphism (Site not responding. Last check: 2007-11-07)
		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)

	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)

	ABSTRACTS DROSTE
		The cofinality of a group G is the cardinality of the length of a shortest chain of proper subgroups terminating at G.
		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.
		In the classes of infinite symmetric groups, their normal subgroups, and their factor groups, we determine those groups which are equivalent in the sense that they may not be distinguished by the solvability of a system of finitely many equations in variables and parameters.
	www.informatik.uni-leipzig.de /~droste/droabal.html (3181 words)

	Group isomorphism (Site not responding. Last check: 2007-11-07)
		In abstract algebra given two groups (G ) and (H @) a group* isomorphism from (G ) to (H @) is a bijective group* homomorphism from G to H.
		The group Z of integers (with addition) is a subgroup of R and the factor group R / Z is isomorphic to the group S
		The composition of two automorphism is an automorphism and with this operation the of all automorphisms of a group G denoted by Aut(G) forms itself a group the automorphism group of G.
	www.freeglossary.com /Group_isomorphism (462 words)

	MA3131 Group Theory
		Therefore it is essential that students are familiar with the basics of group theory (as taught in a second year course covering group theory), although a brief revision of this material will take place at the start of the course.
		This course aims to present the fundamental ideas of group theory by studying the structure theorems and decomposition concepts that arise in attempts to understand groups in terms of less complicated groups.
		These attempts are most successful in studying finite groups because there is a sense in which any finite group can be regarded as a group built from finite simple groups.
	www.mcs.le.ac.uk /Modules/MA-02-03/MA3131.html (691 words)

	Isohedrally compatible tilings (Site not responding. Last check: 2007-11-07)
		Indeed, this is referred to as the group of symmetries, or automorphism group of the tiling.
		It is a surprising fact that there are only 17 different possible automorphism groups arising from periodic tilings, the so-called (plane) crystallographic groups or wallpaper groups.
		Further, since reflections in the automorphism group of a tiling must have their main lines lie on edges of tiles or bisect tiles, there is only a very limited number of unmarked tiles for which there are isohedral tilings that have p4m, pmm or p6m as automorphism groups.
	www.mi.sanu.ac.yu /vismath/maynard (1642 words)

	Introduction (Site not responding. Last check: 2007-11-07)
		An automorphism group A is a group of bijective homomorphisms from some group G to itself.
		The group G is called the base group of A and we say that A acts on G. Each Magma automorphism group A stores, as part of its data structure, a generating set for its base group, and each automorphism in A is described by its action on these generators.
		The objects are the automorphism groups and the morphisms are group homomorphisms.
	www.umich.edu /~gpcc/scs/magma/text407.htm (121 words)

	Automorphism Group and Isometry Testing
		This function computes the automorphism group G of a lattice L which is defined to be the group of those automorphisms of the Z-module underlying L which preserve the inner product of L. L must be an exact lattice (over Z or Q).
		G does not act on the elements of L, since there is no natural matrix action of the automorphism group on L in the case that the rank of L is less than its degree.
		This function computes the subgroup of the automorphism group of L which fixes also the forms given in the set or sequence F. The matrices in F are not required to be positive definite or even symmetric.
	www.math.lsu.edu /magma/text813.htm (1539 words)

	3-D Crystals XVIII
		That all the automorphisms -- regarded as permutations of the elements of the given group -- themselves form a group, is readily appreciated.
		The automorphisms of a group are those one-to-one correspondences between the elements of that group with those of a copy of it that preserve products.
		The set of all possible automorphisms of the group G itself forms a group under successive application of those permutations, and is called the automorphism group of G, i.e.
	home.hetnet.nl /~turing/d3_lattice_18.html (3950 words)

	Automorphism Group of a Graph or Digraph (Site not responding. Last check: 2007-11-07)
		Compute the automorphism group of the graph G. Note that G may be directed or undirected.
		Compute the automorphism group of the graph G according to the parameters parameters.
		If the parameter Stabilizer is assigned a partition P of the vertex-set of G as its value, then the subgroup of the automorphism group of G that preserves the partition P will be computed.
	www.math.ufl.edu /help/magma/text491.html (324 words)

	Outer automorphism group - The Jiggies Reference Guide (Site not responding. Last check: 2007-11-07)
		The outer automorphism group of a group G is the quotient of the automorphism group Aut(G) by its inner automorphism group Inn(G).
		This is a consequence of the fact that quotients of groups are not in general subgroups.
		It was conjectured by Schreier that Out(G) is always a solvable group when G is a finite simple group.
	www.jiggies.com /reference/Outer_automorphism_group (154 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)

	FREE GROUPS
		(F14) Let F be a non-cyclic free group of finite rank, and G a finitely generated residually finite group.
		(F19) (M.Wicks) (a) Let F be a non-cyclic free group of rank n, and P(n,k) the number of its primitive elements of length k.
		Suppose that A, B, H, K are free groups of finite ranks.
	zebra.sci.ccny.cuny.edu /web/nygtc/problems/probfree.html (1478 words)

	Combinatorial Software
		CoCo: developed by Igor Faradzev, Mikhail Klin et al., at the Moscow Institute for System Studies, ported to UNIX by Andries E. Brouwer, is a package for the computation with permutations groups, group actions, association schemes, strongly regular graphs, and related objects.
		Groups and Graphs: is a software package for graphs, digraphs, geometric configurations, combinatorial designs, and their automorphism groups.
		Symmetrica: written in C by Axel Kohnert, is a collection of routines to compute with symmetric functions and Schubert polynomials, ordinary, modular, and projective representations of the symmetric group and Hecke algebras of type A.
	www.mat.univie.ac.at /~slc/divers/software.html (2130 words)

	[No title]
		Also the generators and orders of their automorphism groups are determined.
		Further, this group is the full automorphism group of them.
		In this paper, $N$ will be a topological nearring whose additive group is a Euclidean $N$-group and $G$ will be the additive group $R$ of real numbers.
	www.math.hr /glasnik/vol_31/gl96-1.txt (729 words)

	SIGGS Theory Model: Automorphism (Site not responding. Last check: 2007-11-07)
		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.
		Group projects also allow for automorphism as group members change roles to fulfill changing needs of the project.
	www.indiana.edu /~educr547/frick/automorp.html (213 words)

	Citations: The holomorphic automorphism group of the complex disk - Ungar (ResearchIndex) (Site not responding. Last check: 2007-11-07)
		Abraham A. Ungar, "The holomorphic automorphism group of the complex disk," Aequat.
		The holomorphic automorphism group of the complex disk.
		Involutory automorphisms) An automorphism of a group G is involutory if it equals its inverse automorphism.
	citeseer.ist.psu.edu /context/859333/0 (523 words)

	Domains With Non-Compact Automorphism Group: A Survey (ResearchIndex) (Site not responding. Last check: 2007-11-07)
		We survey results arising from the study of domains in C n with noncompact automorphism group.
		Beginning with a well-known characterization of the unit ball, we develop ideas toward a consideration of weakly pseudoconvex (and even non-pseudoconvex) domains with particular emphasis on characterizations of (i) smoothly bounded domains with non-compact automorphism group and (ii) the Levi geometry of boundary orbit accumulation points.
		1 Characterization of the bidisc by its automorphism group (context) - Wong - 1995
	citeseer.ist.psu.edu /230747.html (1003 words)

	Generating p-groups
		The input to the algorithm is a p-group G. The output is a sequence of p-group generation processes: each process provides access to a power-conjugate presentation for a descendant which satisfies chosen parameters and a description of the automorphism group of the descendant.
		Each process provides access to a pcp for a descendant of G and to a sequence of generators for the automorphism group of this descendant.
		This function extracts the group H defined by the pc-presentation associated with the p-group generation process P and sets up H as a member of the category GrpPC of soluble groups.
	www.math.uiuc.edu /Software/magma/text233.html (1457 words)

	Fischer group
		The Fischer groups are finite groups named after Bernd Fischer, who discovered them while investigating 3-transposition groups.
		The typical example of a 3-transposition group is a symmetric group, where the Fischer transpositions are genuinely transpositions.
		The groups he found fell into several infinite classes (as well as the symmetric groups, certain classes of symplectic and orthogonal groups fulfilled his conditions) with the exception of the three Fischer groups.
	www.fact-index.com /f/fi/fischer_group.html (319 words)

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