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Topic: Inner automorphism

	Inner Automorphisms (Site not responding. Last check: 2007-10-12)
		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)

	tScholars.com \| Inner automorphism (Site not responding. Last check: 2007-10-12)
		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.
	www.tscholars.com /encyclopedia/Inner_automorphism (381 words)

	Automorphisms (Site not responding. Last check: 2007-10-12)
		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.
		Given a group of Lie type automorphism h, this returns a field automorphism f, a graph automorphism g and an inner automorphism i such that h=fgi.
	www.math.lsu.edu /magma/text1053.htm (415 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/InnerAutomorphism.html (102 words)

	Monoids and Groups. Group Theory and Symmetries - Numericana
		Inner automorphisms: Inn(G) is isomorphic to the quotient of G by its center.
		Under function composition, inner automorphisms form a normal subgroup, denoted Inn(G), of the group of the automorphisms on G, denoted Aut(G) (itself a subgroup of Sym(G), the symmetric group on G).
		Automorphic functions (originally dubbed "Fuchsian functions" by Poincaré, around 1884) are meromorphic functions (i.e., ratios of two holomorphic functions; analytic functions of a complex variable) which are invariant under a countable infinity of Möbius transformations.
	home.att.net /~numericana/answer/groups.htm (5181 words)

	GAP Manual: 58 Automorphism Groups of Special Ag Groups (Site not responding. Last check: 2007-10-12)
		Automorphisms are represented by their action on the sag group generating set of the input group.
		The performance of the automorphism group algorithm is highly dependent on the structure of the input group.
		It is followed by a description of automorphism group elements and their operations (see Automorphism Group Elements and Operations for Automorphism Group Elements).
	www.math.jussieu.fr /~jmichel/htm/CHAP058.htm (1231 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.
		As has been explained, the identity automorphism is obtained by transforming each element of the group by the transforming element 1, and we let this identity automorphism correspond with the first row of the above array (depicting the six automorphisms, derived by the interchange of generators).
		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)

	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 (1029 words)

	[No title]
		That fact rarely holds for generalized conjugation, but for conjugation produced by an inner automorphism, we are at least guaranteed that {\it some} element falls into a conjugacy class by itself.
		A purely empirical observation that might be made on the basis of the examples given in Section~\ref{sec-eg} is that when $\phi$ is not an inner automorphism, $\phi$-conjugacy classes tend to be larger in size and fewer in number than ordinary conjugacy classes.
		Throughout this section, $\phi$ and $\psi$ denote automorphisms with the property that $\phi\psi\inv$ is an inner automorphism.
	www.rose-hulman.edu /Users/faculty/sherman/REU96/pramod.txt (4827 words)

	Stored Attributes of an Automorphism Group
		Automorphism groups have several attributes that may be stored as part of their data structure.
		: The order of the outer automorphism group associated with A. It is an integer and may be set by giving either an integer or a factored integer.
		: A sequence of generators of A known to be inner automorphisms.
	www.umich.edu /~gpcc/scs/magma/text413.htm (588 words)

	Automorphisms
		The elements of a group of automorphisms are automorphisms of the base group, so Magma treats them as both homomorphisms and group elements.
		Let A be a group of automorphisms of a group G and let i be an integer such that -n <= i <= n, where n is the number of generators of A. This operator returns the i-th generator for A. A negative subscript indicates that the inverse of the generator is to be created.
		Let A be a group of automorphisms of a group G. Given an automorphism f of G, represented as a Magma map, this function returns the element of A corresponding to f.
	www.math.lsu.edu /magma/text368.htm (504 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)

	Springer Online Reference Works
		» Encyclopaedia of Mathematics » I » Inner automorphism
		Automorphisms that are not inner are called outer automorphisms.
		Other relevant concepts include those of an inner automorphism of a monoid (a semi-group with a unit element) and an inner automorphism of a ring (associative with a unit element), which are introduced in a similar way using invertible elements.
	eom.springer.de /I/i051230.htm (149 words)

	[No title]
		The centre of the group has order 2, and is generated by x^2 = -I. The inner automorphism group is the group modulo its centre, which is PSL(2,Z), and is isomorphic to the free product C_2 * C_3.
		The full automorphism group is at least twice as big as that, because it contains PGL(2,Z), which is the image of integral matrices with determinant 1 or -1.
		Each inner automorphism of G induces the trivial automorphism of G/[G,G], so the inner automorphisms of G lie in ker(phi).
	www.math.niu.edu /~rusin/known-math/01_incoming/SL2Z (938 words)

	Automorphisms and Isomorphisms
		The description is a sequence of matrices which describe the action of each automorphism on the defining generators of the pcp for the class k - 1 p-quotient.
		A generating set for a supplement to the inner automorphism group is returned as a sequence of homomorphisms; each describes the action of a generator of the automorphism group on the generators of the group.
		The function returns a generating set for a supplement to the inner automorphism group of G. This generating set is returned as a sequence of homomorphisms, where each describes the action of an automorphism on the generators of G. [Next] [Prev] [Right] [Left] [Up] [Index] [Root]
	www.math.ufl.edu /help/magma/text249.html (1431 words)

	[No title] (Site not responding. Last check: 2007-10-12)
		None of these automorphisms is inner, as can be seen by looking at the induced actions on the abelianization G/[G,G].
		The last is inverted by the automorphism sending a -> b a^(-1) b -> a and has a cube equal to the automorphism sending a -> a^(-1) c b -> b^(-1) c where c is the commutator a b^(-1) a^(-1) b.
		These three automorphisms together with the inner automorphisms by a and b generate all of Aut(G), and I seem to recall only three generators are really needed; I don't know what sorts of relations bind them.
	www.math.niu.edu /~rusin/known-math/01_incoming/aut_free (592 words)

	Inner automorphism (Site not responding. Last check: 2007-10-12)
		In abstract algebra an inner automorphism of a group is a function f : G
		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).
		This book is a treasure that goes deeper and beyond most if not all of the new thinker's books written in the past several years (if not decades) in the areas of spir...
	www.freeglossary.com /Inner_automorphism (559 words)

	Inner automorphism: Definition and Links by Encyclopedian.com
		In abstract algebra, if G is a group and a is an element of G, then the function f : G
		As the name suggests, f is a group automorphism of G.
		The collection of all inner automorphisms of G forms a normal subgroup of the full automorphism group G.
	www.encyclopedian.com /in/Inner-automorphism.html (186 words)

	Special Homomorphisms and Isomorphisms
		An isomorphism of a group G onto itself is called an automorphism.
		In words, Proposition 7.2.1 says that the conjugate of a subgroup is a subgroup.
		(if there are any) which are not inner automorphisms are called outer automorphisms.
	web.usna.navy.mil /~wdj/tonybook/gpthry/node39.html (199 words)

	[No title]
		If $G\\lhd H$, then conjugation by any\ element $h\\in H$ gives a well-defined (but not necessarily inner)\ automorphism of $G$; the $\\Ad h$ notation will be used in such cases\ as well.\ \ Earlier, the notations $Z^\\phi_x$ and $C^\\phi_x$ were introduced for\ the $\\phi$-centralizer and the $\\phi$-conjugacy class of $x$,\ respectively.
		Thus\ $\\ker^\\infty\\phi$ is a normal subgroup.\ \\end\{proof\}\ \ \\begin\{prop\}\ Given an endomorphism $\\phi: G\\to G$, the induced map\ $\\phi^\{\\ker\}:G/\\ker^\\infty\\phi \\to G/\\ker^\\infty\\phi$ is an automorphism.
		This map clearly preserves the action of\ that generator on $G$; furthermore, it is well defined since powers of\ those generators which are inner automorphisms have been identified\ with the appropriate elements of $G$.\ \\end\{proof\}\ \ \\noindent\ Now, recall that $G$ is a normal subgroup of $\\Eact_\\phi G$.
	www.rose-hulman.edu /users/faculty/sherman/REU96/prmdtrdo.rtf (3183 words)

	Normal Endomorphisms
		An endomorphism f is normal if it commutes with every inner automorphism.
		If g is indecomposable, and acc and dcc, and f is a normal endomorphism, then f is an automorphism or it is nilpotent.
		Suppose it is an automorphism, with an inverse e.
	www.mathreference.com /grp-ks,endo.html (653 words)

	[XMod] 2 2d-objects
		Examples of these are implicitly included in the fourth part, namely inclusions of normal sub-crossed modules, and the inner morphism from a crossed module to its actor.
		An automorphism crossed module has as range a subgroup R of the automorphism group \Aut(S) of S which contains the inner automorphism group of S.
		Here is a simple example of an automorphism crossed module, using a cyclic group of size five.
	www-groups.dcs.st-and.ac.uk /gap/Manuals/pkg/xmod/htm/CHAP002.htm (1187 words)

	Stored Attributes of an Automorphism Group
		Groups of automorphisms have several attributes that may be stored as part of their data structure.
		function returns whether the group of automorphisms A has the attribute named by the string s defined and, if so, also returns the value of the attribute.
		procedure sets the attribute of the group of automorphisms group named by string s to have value v.
	magma.maths.usyd.edu.au /magma/htmlhelp/text387.htm (589 words)

	[No title] (Site not responding. Last check: 2007-10-12)
		] Ice composing the inner portion of a continental glacier or large ice sheet; applied particularly to Greenland ice.
		] The part of a harbor more remote from the sea, as contrasted with the outer harbor; this expression is normally used only in a harbor that is clearly divided into parts, by a narrow passageway or artificial structure; the inner harbor generally has additional protection and is often the principal berthing area.
		The inner product of two tensors is the contracted tensor obtained from their product by means of pairing contravariant indices of one with covariant indices of the other.
	www.accessscience.com /Dictionary/I/I11/DictI11.html (1870 words)

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