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Topic: Group isomorphism

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	Group isomorphism - Definition, explanation
		In abstract algebra, a group isomorphism is a function between two groupss that sets up a one-to-one correspondence between the elements of the groups in a way that respects the given group operations.
		Given two groups (G, ) and (H, @), a group* isomorphism from (G, ) to (H, @) is a bijective group* homomorphism from G to H.
		An isomorphism from a group G to G itself is called an automorphism of G.
	www.calsky.com /lexikon/en/txt/g/gr/group_isomorphism.php (525 words)

	Science Fair Projects - Group isomorphism
		In abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in a way that respects the given group operations.
		Given two groups (G, ) and (H, @), a group* isomorphism from (G, ) to (H, @) is a bijective group* homomorphism from G to H.
		An isomorphism from a group G to G itself is called an automorphism of G.
	www.all-science-fair-projects.com /science_fair_projects_encyclopedia/Group_isomorphism (535 words)

	Kids.Net.Au - Encyclopedia > Group isomorphism (Site not responding. Last check: 2007-11-07)
		Given two mathematical 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
		An isomorphism from a group G to G is called an automorphism of G.
	www.kids.net.au /encyclopedia-wiki/gr/Group_isomorphism (311 words)

	PlanetMath: fundamental group (Site not responding. Last check: 2007-11-07)
		In particular, this means that the fundamental group of a (non-empty) path-connected space is well-defined, up to isomorphism, without the need to specify a basepoint.
		Homotopy groups generalize the concept of the fundamental group to higher dimensions.
		This is version 12 of fundamental group, born on 2001-11-14, modified 2006-10-07.
	planetmath.org /encyclopedia/FundamentalGroup.html (254 words)

	Applied Group Theory
		Group theory is that branch of mathematics concerned with the study of groups.
		It is the preimage of the identity in the codomain of a group homomorphism.
		Isomorphic groups can be thought of as essentially the same, only with different labels on the individual elements.
	applied-group.com.ru (508 words)

	PlanetMath: Lie group (Site not responding. Last check: 2007-11-07)
		A Lie group is a group endowed with a compatible analytic structure.
		Thus, a homomorphism in the category of Lie groups is a group homomorphism that is simultaneously an analytic mapping between two real-analytic manifolds.
		The name “Lie group” honours the Norwegian mathematician Sophus Lie who pioneered and developed the theory of continuous transformation groups and the corresponding theory of Lie algebras of vector fields (the group's infinitesimal generators, as Lie termed them).
	planetmath.org /encyclopedia/LieGroup.html (657 words)

	Group isomorphism: Definition and Links by Encyclopedian.com
		...Group isomorphism Group isomorphism Given two mathematical groups (G, ) and...) and (H, @) a group isomorphism from (G, ) to (H, @) is a bijective group* homomorphism...to (H, @) is a bijective group homomorphism from G to H.
		...normal subgroup is the kernel of a group isomorphism and vice versa.
		Two groups are isomorphic if there exists an group isomorphism mapping...
	www.encyclopedian.com /gr/Group-isomorphism.html (484 words)

	[No title]
		Group theory is a powerful method for analyzing abstract and physical systems in which symmetry --the intrinsic property of an object to remain invariant under certain classes of transformations-- is present because the mathematical study of symmetry is systematized and formalized in group theory.
		group +------------------------------------------------------------ A group is an object consisting of a set G and a law of composition (or binary operation) L on G satisfying: L is associative.
		A group that is not trivial is nontrivial.
	abel.math.harvard.edu /~knill/sofia/data/group.txt (4457 words)

	physics - Group isomorphism
		In abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in a way that respects the given group operations.
		Given two groups (G, ) and (H, @), a group* isomorphism from (G, ) to (H, @) is a bijective group* homomorphism from G to H.
		An isomorphism from a group G to G itself is called an automorphism of G.
	www.physicsdaily.com /physics/Group_isomorphism (384 words)

	PlanetMath: group homomorphism (Site not responding. Last check: 2007-11-07)
		A composition of group homomorphisms is again a homomorphism.
		are isomorphic, meaning they are basically the same group (have the same structure).
		This is version 22 of group homomorphism, born on 2001-11-08, modified 2006-10-16.
	planetmath.org /encyclopedia/GroupHomomorphism.html (157 words)

	Springer Online Reference Works (Site not responding. Last check: 2007-11-07)
		A quasi-cyclic group is the union of an ascending chain of cyclic groups
		Divisible group), and each divisible Abelian group is the direct sum of a set of groups that are isomorphic to the additive group of rational numbers and to quasi-cyclic groups for certain prime numbers
		A quasi-cyclic group coincides with its Frattini subgroup.
	eom.springer.de /Q/q076440.htm (294 words)

	The Dispatch - Serving the Lexington, NC - News (Site not responding. Last check: 2007-11-07)
		Informally, an isomorphism is a kind of mapping between objects, which shows a relationship between two properties or operations.
		isomorphism" applies when two complex structures can be mapped onto each other, in such a way that to each part of one structure there is a corresponding part in the other structure, where "corresponding" means that the two parts play similar roles in their respective structures.
		Isomorphisms are studied in mathematics in order to extend insights from one phenomenon to others: if two objects are isomorphic, then any property which is preserved by an isomorphism and which is true of one of the objects is also true of the other.
	www.the-dispatch.com /apps/pbcs.dll/section?category=NEWS&template=wiki&text=isomorphism (852 words)

	Automorphism Information
		Inverses: by definition every isomorphism has an inverse which is also an isomorphism, and since the inverse is also an endomorphism of the same object it is an automorphism.
		A group automorphism is a group isomorphism from a group to itself.
		The inner automorphisms are the conjugations by the elements of the group itself.
	www.bookrags.com /wiki/Automorphism (864 words)

	The Discontinuous Groups of Rotation and Translation in the Plane
		Group category 2.2.2.1: Contain non-parallel translations and rotations where the least positive angle is 2*π/2.
		Group category 2.2.2.2: Contain non-parallel translations and rotations where the least positive angle is 2*π/3.
		Group category 2.2.2.5: Contain non-parallel translations and rotations where the least positive angle is not one of 2*π/n with n = {2,3,4,6}.
	xahlee.org /Wallpaper_dir/c0_WallPaper.html (155 words)

	Standard Math Facts
		A group G has an operation (written additively or multiplicatively) mapping GxG to G. There is an identity element e, an inverse for each element in the group and the operation is associative.
		A subgroup is a subset of a group which is a group under the inherited operation.
		Thus f(G), the image of G in H is a subgroup of H. The kernal of f, or ker(f) is all elements x of G with f(x) = identity in H. A group isomorphism is a homomorphism which is one-to-one and onto.
	www.jt-actuary.com /mathfacts.htm (538 words)

	The Dispatch - Serving the Lexington, NC - News (Site not responding. Last check: 2007-11-07)
		In mathematics, a topological group is a group G together with a topology on G such that the group and topological structures are compatible.
		An isomorphism of topological groups is a group isomorphism which is also a homeomorphism of the underlying topological spaces.
		An example of a topological group which is not a Lie group is given by the rational numbers Q with the topology inherited from R.
	www.the-dispatch.com /apps/pbcs.dll/section?category=NEWS&template=wiki&text=topological_group (1067 words)

	[No title]
		is_isomorphism This symbol is a boolean function with three arguments.
		right_inverse_multiplication This symbol is a function with two arguments, which should be a group M and an element x of M. When applied to M and x, it denotes right multiplication on M by the inverse of x.
		-1 conjugation This symbol is a function with two arguments, which should be a group M and an element x of M. When applied to M and x, it denotes conjugation on M by x.
	www.win.tue.nl /~amc/oz/om/cds/group2.html (161 words)

	ABSTRACT ALGEBRA ON LINE: Groups (Site not responding. Last check: 2007-11-07)
		A group G is said to be a finite group if the set G has a finite number of elements.
		Let G be a group, and let H be a subset of G. Then H is called a subgroup of G if H is itself a group, under the operation induced by G. Proposition.
		Any subgroup of the symmetric group Sym(S) on a set S is called a permutation group or group of permutations.
	www.math.niu.edu /~beachy/aaol/groups.html (1115 words)

	Cyclic Groups and Subgroups
		Groups that can be generated in their entirety from one member are called cyclic groups.
		Thus for a cyclic group we have the definition that all the elements may be generated from a single element together with its inverse.
		All infinite cyclic groups are isomorphic to the additive group of integers.
	members.tripod.com /~dogschool/cyclic.html (2281 words)

	Finite Groups (Site not responding. Last check: 2007-11-07)
		A function f from a group (G,) to a group* (H,) is said to be a group* homomorphism if for all a and b in G we have f(ab) = f(a)f(b).
		If f is one-to-one and onto and a group homomorphism, then it is said to be a group isomorphism.
		If f is a group isomorphism and G = H, then f is said to be a group automorphism.
	diamond.boisestate.edu /~liljanab/crypto2s06/finite_groups.htm (671 words)

	Group
		Group of galaxies, a small number of galaxies near each other; a special case is the Local Group, containing the Milky Way Galaxy
		Group (sociology), a sub-set of a culture or of a society
		Functional group, a functional entity consisting of certain atoms whose presence provides a certain property to a molecule.
	www.brainyencyclopedia.com /encyclopedia/g/gr/group.html (207 words)

	Exponential Partitions
		For example, with p = 5 we have the additive group modulo 4: 0 1 2 3 0 0 1 2 3 1 1 2 3 0 2 2 3 0 1 3 3 0 1 2 Notice that there are isomorphisms between this group and the previous multiplicative group.
		Notice that, in the multiplicative group, the operation x -> x^k can be performed by starting with the identity element (i.e., 1) and multiplying by x a total of k times.
		In the isomorphic additive group this corresponds to starting with the identity element (i.e., 0) and adding the number y that maps to x a total of k times.
	www.mathpages.com /home/kmath264.htm (794 words)

	Good Math has moved to ScienceBlogs: Group Isomorphism: Defining ...
		A homomorphism between two groups A and B is a function that maps members of A to members of B in a way that preserves the group property.
		A group isomorphism is a group homomorphism f, where f is a bijection.
		The symmetry transformation of the pentagon is probably best seen as either the group applied to a set, or as a piece of a group permutation, rather than as a complete group permutation.
	goodmath.blogspot.com /2006/04/group-isomorphism-defining-symmetry.html (796 words)

	Isomorphisms and Transformations
		A hyperelliptic curve isomorphism curve defined by the data of a linear fractional transformation t(x:z) = (ax + bz:cx + dz), a scale factor e, and a polynomial u(x) of degree at most g + 1, where g is the genus of the curve.
		An isomorphism can be created from the parent structures by coercing a tuple < [a, b, c, d], e, u > into the structure of isomorphisms between two hyperelliptic curves, or by creating it as a transformation of a given curve, i.e.
		So in the former case the automorphism group is an extension of the elliptic curve automorphism group (of order 2) by Z/2Z, and in latter case the automorphism group is an extension (of the group of order 12) by the abelian group isomorphic to Z/2Z x Z/2Z.
	www.umich.edu /~gpcc/scs/magma/text1076.htm (1190 words)

	:::► Dictionary of Meaning www.dictionary-of-meaning.com ◄:::
		The first isomorphism theorem for vector spaces states that this quotient space is naturally isomorphic to the image (function) of T (which is a subspace of W).
		The first isomorphism theorem for groups states that this quotient group is naturally isomorphic to the image (function) of f (which is a subgroup of H).
		The first isomorphism theorem in general universal algebra states that this quotient algebra is naturally isomorphic to the image (function) of f (which is a image (function) of f (which is a subalgebra of B).
	www.dictionary-of-meaning.com /kernel_(algebra).html (2117 words)

	Exercises 4
		are isomorphic as abelian groups but not as rings (under, of course, the usual addition and multiplication).
		are isomorphic both as groups and as rings.
		If the additive group of a ring is cyclic, generated (say) by an element a, prove that the multiplication is determined once you know a.
	www-groups.dcs.st-and.ac.uk /~john/MT4517/Tutorials/T4.html (253 words)

	ABSTRACT ALGEBRA: OnLine Study Guide, Section 3.4 (Site not responding. Last check: 2007-11-07)
		But if we ask for a list of abelian groups of order 8 that comes with a guarantee that any possible abelian group of order 8 must be isomorphic to one of the groups on the list, then the question becomes manageable.
		If you can show that one group has a property that the other one does not have, then you can decide that two groups are not isomorphic (provided that the property would have been transferred by any isomorphism).
		The analysis of groups of order 4 given on pages 103 and 104 of the text shows that if G = 4, then either G has at least one element of order 4, and is therefore cyclic, or G has three elements of order 2 (not counting the identity).
	www.math.niu.edu /~beachy/abstract_algebra/study_guide/34.html (1037 words)

	Group Theory & Rubik's Cube
		Group theory is the study of the algebra of transformations and symmetry.
		Given an element x of a group G, the orbit of x is the set of all elements of G which are generated by x, i.e.
		A representation of a group G is a set of matrices M which are homomorphic to the group.
	akbar.marlboro.edu /~mahoney/courses/Spr00/rubik.html (3602 words)

	[No title]
		If G is a group, then it is a group-homomorphism.
		groups G aint what I was meaning.....rather it is the mapping f which
		Nowhere is it implied that for an isomorphism
	www.mathforum.org /kb/plaintext.jspa?messageID=5055044 (699 words)

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