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Topic: Multiplicative group

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	PlanetMath: groups in field
		This is, apparently, isomorphic to the additive group
		is not 2 and denote the multiplicative group of
		This is version 18 of groups in field, born on 2004-10-06, modified 2006-08-21.
	planetmath.org /encyclopedia/AdditiveGroupOfTheField.html (202 words)

	PlanetMath: examples of groups (Site not responding. Last check: 2007-10-17)
		More generally, any (skew) field gives rise to two groups: the additive group of all field elements with 0 as identity element, and the multiplicative group of all non-zero field elements with 1 as identity element.
		This is the automorphism group of the given object and captures its “internal symmetries”.
		In Galois theory, the symmetry groups of field extensions (or equivalently: the symmetry groups of solutions to polynomial equations) are the central object of study; they are called Galois groups.
	planetmath.org /encyclopedia/ExamplesOfGroups.html (1011 words)

	Monoids and Groups. Group Theory and Symmetries - Numericana
		The centralizer in a group G of a subset E consists of all the elements of G which commute with every element of E. It is a subgroup of G. The centralizer in G of G itself is the center of G (it's the intersection of all centralizers in G).
		The alternating group is the derived subgroup of the symmetric group: A
		The derived subgroup of the Quaternion group is {+1,-1}.
	home.att.net /~numericana/answer/groups.htm (4881 words)

	Springer Online Reference Works (Site not responding. Last check: 2007-10-17)
		This group is isomorphic to the multiplicative group of all roots of the equations
		A quasi-cyclic group is the union of an ascending chain of cyclic groups
		A quasi-cyclic group coincides with its Frattini subgroup.
	eom.springer.de /Q/q076440.htm (294 words)

	Béla Bódi (Adalbert Bovdi)
		[37] A. Bovdi, Unitary subgroup of the multiplicative group of group algebras, Visnik Kiew University, 27 (1985), 17-18.
		[39] A. Bovdi and I. Khripta, Group algebras with polycyclic multiplicative group, Ukrain.
		[47] A. Bovdi and A. akács, Unitary subgroup of the multiplicative group of a modular group algebra of a finite abelian p-group, Mat.Zametki, 45 (1989), N6, 23-29.
	www.math.klte.hu /algebra/bodiba.htm (1411 words)

	Multiplicative Group -- from Wolfram MathWorld
		A group whose group operation is identified with multiplication.
		In a multiplicative group, the identity element is denoted 1, and the inverse of the element
		This is the cases of transformation groups (such as the rotation group) and the symmetric groups and their subgroups (such as the alternating groups).
	mathworld.wolfram.com /MultiplicativeGroup.html (291 words)

	Group (mathematics) - Wikipedia, the free encyclopedia
		A group G is said to be an abelian group (or commutative) if the operation is commutative, that is, for all a, b in G, a * b = b * a.
		A cyclic group is a group that is generated by a single element.
		In multiplicative groups, the identity element is denoted by 1.
	en.wikipedia.org /wiki/Group_(mathematics) (2628 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)

	Multiplicative group - Wikipedia, the free encyclopedia
		See also: multiplicative group of integers modulo n, additive group.
		The group scheme of n-th roots of unity is by definition the kernel of the n-power map on the multiplicative group, considered as a group scheme.
		The Galois cohomology of this group scheme is a way of expressing Kummer theory.
	en.wikipedia.org /wiki/Multiplicative_group (278 words)

	ABSTRACT ALGEBRA ON LINE: Groups
		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)

	Springer Online Reference Works (Site not responding. Last check: 2007-10-17)
		The quaternion group is a Hamilton group, and the minimal Hamilton group in the sense that any non-Abelian Hamilton group contains a subgroup isomorphic to the quaternion group.
		The intersection of all non-trivial subgroups of the quaternion group (and also of any generalized quaternion group) is a non-trivial subgroup.
		Sometimes the term "quaternion group" is used to denote various subgroups of the multiplicative group of the algebra of quaternions and related topological groups.
	eom.springer.de /q/q076780.htm (215 words)

	Groups of Transformations in Music
		satisfy the group axioms, where the binary operation is addition, the identity for addition is 0, and the inverse of "a" is "-a".
		To see this group as a group of translations is easy, it is a group of discrete translations of the set of integers it self.
		The table on the left is an abstraction for the multiplicative group of {1, -1, i, -i}, where i is the square root of (-1).
	graham.main.nc.us /~bhammel/MUSIC/group.html (1521 words)

	3.0 The group Zp*
		Multiplication ends by taking the remainder on division by p; this ensures closure.
		Each number x in a multiplicative group has a multiplicative inverse element in the group; that is an integer x^{-1} such that x x^{-1}= 1 in the group.
		It is possible to define multiplication on the numbers from 1 to 14, always finishing with reduction modulo 15.
	www.certicom.com /index.php?action=ecc_tutorial,math3 (248 words)

	How the FFT constants were found
		Furthermore, it is the case that the multiplicative group of nonzero elements in any finite field is cyclic.
		Thus, if m is prime, the multiplicative group of nonzero elements is a cyclic group of order m-1 containing m-1.
		Assuming m is prime, the multiplicative group must have a generator.
	www.cis.ksu.edu /~rhowell/calculator/how.html (515 words)

	Finite fields
		The multiplicative group of a finite field is cyclic.
		Consider the multiplicative group of the field with 9 elements.
		So the multiplicative group is a direct product of cyclic groups corresponding to various primes and hence is cyclic.
	www-groups.dcs.st-and.ac.uk /~john/MT4517/Lectures/L9.html (909 words)

	Scheme for arithmetic operations in finite field and group operations over elliptic curves realizing improved ...
		The method of claim 6, wherein the calculating steps calculate a natural number multiple (x.sub.3, y.sub.3)=n(x.sub.1, y.sub.1) of an element (x.sub.1, y.sub.1) of a group over elliptic curves, where n is a natural number, as a combination of multiplicative inverse calculations, additions, and double calculations in x.sub.1 and y.sub.1.
		This n bit multiplication unit 500 is for a case where one of multiplying number (x.sub.1 =a) is constant, as in the n bit multiplication unit 111 of FIG.
		Consequently, by iterating the reduction of multiplication into multiplication in subfield, it becomes possible to utilize a subfield that has a parameter n for which a table size can be reduced to that of a fast read accessible memory (cache memory).
	www.freepatentsonline.com /6202076.html (11426 words)

	c6s3p4df1ex (Site not responding. Last check: 2007-10-17)
		The multiplicative groups of Q, R and C.
		Suppose p is a prime, then multiplication defines a group on Z/pZ\{0}.
		Multiplication does not define a group structure on Q[X], since X has no inverse.
	www.win.tue.nl /~ida/alge/c6s3p4df1ex.html (176 words)

	Cryptosystems Based on Discrete Logarithms
		It is well known that the multiplicative group of nonzero elements of
		In practice, one has to be extremely careful in choosing the group G so that the discrete logarithm problem is hard.
		Note that in the ElGamal cryptosystem, it is required that a plaintext be in the group.
	www.math.clemson.edu /faculty/Gao/crypto_mod/node4.html (1067 words)

	Multiplicative Inverse -- from Wolfram MathWorld
		In a monoid or multiplicative group where the operation is a product
		To detect the multiplicative inverse of a given element in the multiplication table of finite multiplicative group, traverse the element's row until the identity element 1 is encountered, and then go up to the top row.
		SEE ALSO: Additive Inverse, Invertible Element, Multiplicative Identity, Multiplicative Group.
	mathworld.wolfram.com /MultiplicativeInverse.html (139 words)

	[ALNUTH] 2 Methods for number fields
		However, it is a non-trivial task to provide a presentation for this abelian group.
		The most useful representation for such groups is as pcp group.
		Determine a pcp presentation for the multiplicative group of
	www-groups.dcs.st-and.ac.uk /~gap/Manuals/pkg/alnuth/htm/CHAP002.htm (635 words)

	Cyclic Multiplicative Groups (Site not responding. Last check: 2007-10-17)
		Every finite multiplicative subgroup of a field is cyclic.
		Let g be a multiplicative subgroup of a field, with g
		Given a finite field, let b generate the multiplicative group for the field.
	www.mathreference.com /fld-fin,cmg.html (98 words)

	Abelian groups
		is a group which is not necessarily commutative then we call
		Example 5.6.2 The integers, with ordinary addition as the group operation, is an abelian group.
		We have seen (multiplicative) cyclic groups before in our discussion of the discrete log problem.
	web.usna.navy.mil /~wdj/book/node173.html (149 words)

	multiplicative groups
		Letting F be a finite field, how would one show that the multiplicative group must be cyclic?
		I know that if the order of F = n, then the multiplicative group (say, F*) has order n - 1 = m.
		google for multiplicative group finite field cyclic, and look at the first hit from planetmath
	www.physicsforums.com /showthread.php?t=64800 (139 words)

	sci.crypt: Re: multiplicative group question (Site not responding. Last check: 2007-10-17)
		Next in thread: Kristian Gjøsteen: "Re: multiplicative group question"
		You might enjoy reading Susan Hohenberger's master's thesis.
		She investigates the notion of groups that satisfy 2) plus a number
	www.derkeiler.com /Newsgroups/sci.crypt/2005-06/2131.html (212 words)

	Finding the order of the multiplicative group (mod n)...zuh? (kottke.org)
		Finding the order of the multiplicative group (mod n)...zuh?
		Weblogs are usually pretty easy for readers to get into.
		If you've reached this point by accident, I suggest panic.
	www.kottke.org /02/09/order-multiplicative-group (480 words)

	[No title]
		= {1,2,4} = H¡H¿ çÿ{$ª!ðpB ðX ÃðH¿ÀËpÆÑÒÓÕÿ?ðppðpð· ðX ãðT‘”«gÖ³«gÖ³¿¿Àÿ?ðÖ 9ö ð3¨2 generates the subgroup H of GðH ðX ð0“”Þ½h¿ÿ ?ð ÿÿÿ———Ì33ÌÌÌÿ²²²îQïÙ Ú% íðåpð\ð}ð( ð9ö ð\ðÌ ð\ ãðTÄ‘”«gÖ³«gÖ³¿¿Àÿ?ð°ÐP ðH¨Generated Groups¡,Ìfþð ð\ ãðT’”«gÖ³«gÖ³¿¿Àÿ?ðæV t ð¨Exampleð¥ ð\ ãðTä’”«gÖ³«gÖ³¿¿Àÿ?ðÖ¦¾ ð!¨Let G = Z7 = {1,2,3,4,5,6} = the multiplicative* group modulo 7 Let S = {3}.
		= {1,3,2,6,4,5} = G¡H çÿRª!lðpB ð\ ÃðH¿ÀËpÆÑÒÓÕÿ?ðppðpð¥ ð\ ãðTD“”«gÖ³«gÖ³¿¿Àÿ?ðÖ ö ð!¨ 3 generates GðH ð\ ð0“”Þ½h¿ÿ ?ð ÿÿÿ———Ì33ÌÌÌÿ²²²î»ï Ù Ú% WðOð`ðçð( ðX ° ð`ðº ð` ðf””«gÖ³«gÖ³¿¿ÿ ?ð°ÐPðÃ ” ð ðí ð` ãðTd””«gÖ³«gÖ³¿¿Àÿ?ð½t ði¨7For every group* G and element x of G, order(x) =
		Then z = xu = ywv-1u must be an element of yH because w,v-1, and u are elements of H. This shows that xH is a subset of yH.
	www.cs.unm.edu /~gemmell/491/ntheory2.ppt (472 words)

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