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	Dicyclic group - Wikipedia, the free encyclopedia
		There is a superficial resemblance between the dicyclic groups and dihedral groups; both are a sort of "mirroring" of an underlying cyclic group.
		There is a natural 2-to-1 homomorphism from the group of unit quaternions to the 3-dimensional rotation group described at quaternions and spatial rotations.
		Since for a cyclic group of even order, there is always a unique element of order 2, we can see that dicyclic groups are just a specific type of generalized dicyclic group.
	en.wikipedia.org /wiki/Dicyclic_group (494 words)

	Silver halide photographic material and method for forming high contrast negative image using the same - Patent 4681836
		Further, X represents a group having the bonding unit of ##STR1## a group having the bonding unit of ##STR2## a group represented by ##STR3## a heterocyclic ring residue, an aralkyl group (in the case of n=1), or an aryl group substituted by an alkyl group.
		Of groups represented by R.sup.2 in formula (I), the aryl group, which may be substituted, is preferably a monocyclic or dicyclic group, such as a benzene ring or a naphthalene ring, and most preferably a benzene ring.
		Groups preferred as R.sup.2, when G represents a carbonyl group, are a hydrogen atom, a methyl group, a methoxy group, an ethoxy group, and a substituted or unsubstituted phenyl group.
	www.freepatentsonline.com /4681836.html (5499 words)

	Dicyclic group -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-09)
		More generally, when n is a power of 2, the dicyclic group is isomorphic to the (Click link for more info and facts about generalized quaternion group) generalized quaternion group.
		There is a superficial resemblance between the dicyclic groups and (Click link for more info and facts about dihedral group) dihedral groups; both are a sort of "mirroring" of an underlying cyclic group.
		Generalized dicyclic groups, in turn, are examples of (Click link for more info and facts about cyclic extension) cyclic extensions.
	www.absoluteastronomy.com /encyclopedia/d/di/dicyclic_group.htm (609 words)

	Silver halide photographic material - Patent 4824764
		Preferred amino group represented by R.sub.2 are a diethylamino group, a di(2-hydroxyethyl)amino group, a morpholine-1-yl group, a pyridine-1-yl group, imidazol-1-yl group, and a phenylamino group.
		In the case where G is a sulfonyl group, preferred examples of the groups represented by R.sub.2 include a methyl group, an ethyl group, a phenyl group, a 4-methylphenyl group, an o-hydroxybenzyl group and a 2-acetylethyl group.
		The substituents for the group represented by R.sub.4 include a hydroxy group, an alkoxy group having 1 to 9 carbon atoms, a sulfonamide group having 0 to 9 carbon atoms, a carbonamide group having 1 to 9 carbon atoms, an ureido group having 1 to 9 carbon atoms and a group having a positive.sigma.
	www.freepatentsonline.com /4824764.html (6352 words)

	cyclic group
		In mathematics, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element a (called a "generator" of the group) such that all elements of the group are powers of a.
		Similarly, the endomorphism ring of the infinite cyclic group is isomorphic to the ring Z, and its automorphism group is isomorphic to the group of units of the ring Z, i.e.
		The Galois group of every finite field extension of a finite field is finite and cyclic; conversely, given a finite field F and a finite cyclic group G, there is a finite field extension of F whose Galois group is G.
	www.fact-library.com /cyclic_group.html (736 words)

	Dicyclic group (Site not responding. Last check: 2007-10-09)
		In group theory, a dicyclic group is a member of a classof groups which are formed by an extension of a group (generally a cyclic group) by a cyclic group of order 2 (the latter giving the name di-cyclic).
		There is a superficial resemblance between the dicyclic groups and dihedral groups ; both are a sort of "mirroring" of an underlying cyclic group.
		Since for a cyclic group of even order, there is always a unique element of order 2, we can see that dicyclic groups are justa specific type of generalized dicyclic group.
	www.therfcc.org /dicyclic-group-210804.html (323 words)

	PlanetMath: groups of small order
		Below is a list of all possible groups per order up to isomorphism.
		All groups of prime order are isomorphic to a cyclic group of that order.
		"groups of small order" is owned by Daume.
	www.planetmath.org /encyclopedia/GroupsOfSmallOrder.html (160 words)

	Home Fresh : Article 'Permutation group' (Site not responding. Last check: 2007-10-09)
		In mathematics, a permutation group is a group G whose elements are permutations of a given set M, and whose group operation is the composition of permutations in G (which are thought of as bijective functions from the set M to itself); the relationship is often written as (G,M).
		Note that the group of all permutations of a set is the symmetric group; the term permutation group is usually restricted to mean a subgroup of the symmetric group.
		The application of a permutation group to the elements being permuted is called its group action; it has applications in both the study of symmetries, combinatorics and many other branches of mathematics.
	www.home-fresh.net /DisplayArticle49143.html (269 words)

	Quaternion group (Site not responding. Last check: 2007-10-09)
		In group theory, the quaternion group is a non- abelian group of order 8 with a number of interesting properties.
		Note that the resulting group is non- commutative ; for example ij = - ji.
		These groups are members of the still larger family of dicyclic group s.
	www.serebella.com /encyclopedia/article-Quaternion_group.html (595 words)

	Quaternion Group Encyclopedia Article, Definition, History, Biography (Site not responding. Last check: 2007-10-09)
		The inner automorphism group of Q is isomorphic to Q modulo its center, and is therefore also isomorphic to the Klein four-group.
		The quaternion group Q may be regarded as acting on the eight nonzero elements of the 2-dimensional vector space over the finite field GF(3).
		The generalized quaternion groups are members of the still larger family of dicyclic groups.
	www.folkartmuseum.com /encyclopedia/Quaternion_group (674 words)

	GROUP-LAB Overview
		At start-up a default group, the octohedral group, isomorphic to the symmetric group S4, is initialized.
		User-defined groups may be introduced almost at will by specifying an appropriate set of relations in up to four generators, in a manner to be illustrated later in this demo.
		This group is generated by the elements x and y satisfying the relations xxxxxxxx=1, yyXXXX=1, and yxYx=1, where uppercase letters denote the inverses of the corresponding lowercase letters.
	www.math.vt.edu /people/layman/software/overview.htm (1701 words)

	Dicyclic group (Site not responding. Last check: 2007-10-09)
		In group theory a dicyclic group is a member of a class groups which are formed by an extension of a group (generally a cyclic group) by a cyclic group of order (the latter giving the name di-cyclic).
		There is a superficial resemblance between the groups and dihedral groups ; both are a sort of "mirroring" an underlying cyclic group.
		Since for a cyclic group of even there is always a unique element of 2 we can see that dicyclic groups just a specific type of generalized dicyclic
	www.freeglossary.com /Dicyclic_group (811 words)

	List of small groups - Wikpedia (Site not responding. Last check: 2007-10-09)
		The list can be used to determine which known group a given finite group G is isomorphic to: first determine the order of G, then look up the candidates for that order in the list below.
		To distinguish between the remaining candidates, look at the orders of your group's elements, and match it with the orders of the candidate group's elements.
		The group theoretical computer algebra system GAP contains the "Small Groups library" which provides access to descriptions of the groups of "small" order.
	www.bostoncoop.net /~tpryor/wiki/index.php?title=Trivial_group (326 words)

	Group Theory - Groups Up To Order Eight (Site not responding. Last check: 2007-10-09)
		Recall groups of prime order are cyclic, so we need only focus on the cases
		We shall see later that this is indeed a group (associativity turns out to hold) because it is the symmetric group of degree 3 (which is isomorphic to the dihedral group of order 6).
		The quaternion group is a special case of a dicyclic group, groups of order
	rooster.stanford.edu /~ben/maths/group/ordereight.php (363 words)

	Dave Witte Morris' papers in graph theory
		For any Cayley graph on any finite abelian group, this paper determines precisely which elements of the cycle space can be written as sums of hamiltonian cycles.
		The hyperbolic symmetry groups [p, q], [p, q]+, and [p+,q] have certain natural generating sets; this paper determines whether or not the corresponding Cayley digraphs have one-way-infinite or two-way-infinite hamiltonian paths.
		We explicitly determine all of the transitive groups of degree p-squared, p a prime, whose Sylow p-subgroup is not the wreath product of two cyclic groups of order p.
	people.uleth.ca /~dave.morris/GraphTheory.shtml (1150 words)

	Citations: Uber endliche Fastkorper - Zassenhaus (ResearchIndex) (Site not responding. Last check: 2007-10-09)
		The ane groups over the Galois eld and over the Dickson near eld of order p are normal subgroups of index 2 in the group AL(1; p) which is also elusive.
		Then jGj Y a2A (n Gamma a) 18 Groups meeting this bound are called sharp groups.
		In doing so he also gave a full description of those finite groups which have a representation D (over C) such that D(g) has no eigenvalue 1 for any nontrivial element g 2 G. In our terminology such a group is called semiregular.
	citeseer.ist.psu.edu /context/508998/0 (954 words)

	[No title]
		A unitary group code of length n is defined for n >= t as the collection of the matrices of form {DG} where G is from a group of nxn unitary matrices and D is a txn matrix such that DG is a matrix whose elements are drawn from a particular constellation.
		This group code formulation can be used as space-time block codes when perfect CSI is available and also, it can be used to generate a differential space-time modulation (DSTM) scheme.
		In Appendices B-E, it is shown that every full diversity unitary group code with cardinality M and n=t=2 is equivalent to an (M,k) cyclic code or a dicyclic code.
	www.ece.osu.edu /ips/links/hughes.txt (794 words)

	GAP Manual: 48.12. CharTable
		The columns of the table will be sorted in the same order, as the classes of the group, thus allowing a bijection between group and table.
		The computation of character tables needs to identify the classes of group elements very often, so it can be helpful to store a class list of all group elements.
		for the Sylow 2 subgroup of the alternating group A_(11).
	www.math.uiuc.edu /Software/GAP-Manual/CharTable.html (881 words)

	[ref] 71.3 Generic Character Tables
		Generic character tables provide a means for writing down the character tables of all groups in a (usually infinite) series of similar groups, e.g., cyclic groups, or symmetric groups, or the general linear groups
		While the numbers of conjugacy classes for the members of a series of groups are usually not bounded, there is always a fixed finite number of types (equivalence classes) of conjugacy classes; very often the equivalence relation is isomorphism of the centralizers of the representatives.
		For example, the classes of symmetric groups can be parametrized by partitions, corresponding to the cycle structures of permutations.
	wwwmaths.anu.edu.au /research.groups/aat/GAP/www/Manual4/ref/C071S003.htm (827 words)

	CONK! Encyclopedia: List_of_group_theory_topics (Site not responding. Last check: 2007-10-09)
		Encyclopedia page, to provide an overview of the topic and to allow those interested to use the "Related changes" feature to keep abreast of edits in this area.
		See also: List of abstract algebra topics, List of category theory topics, list of Lie group topics.
		Mathematical objects which have (or make use of) a group operation
	www.conk.com /search/encyclopedia.cgi?q=List_of_group_theory_topics (131 words)

	Groups of order < 25
		<12,3>: The alternating group of degree 4: A
		There are 5 isomorphism classes of groups of order 12.
		There are 14 isomorphism classes of groups of order 16.
	web.usna.navy.mil /~wdj/gap/small_groups.html (146 words)

	[CTblLib] 2 The GAP Character Table Library
		Note that class fusions stored on library tables are not guaranteed to be compatible for any two subgroups of a group and their intersection, and they are not guaranteed to be consistent w.r.t.
		Trivial multiplier or outer automorphism group are denoted by an empty string.
		Generic character tables provide a means for writing down the character tables of all groups in a (usually infinite) series of similar groups, e.g., cyclic groups, or symmetric groups, or the general linear groups GL(2,q) where q ranges over certain prime powers.
	www-gap.dcs.st-and.ac.uk /oldsite/pkg/ctbllib/htm/CHAP002.htm (6959 words)

	ORIGIN OF STANDARDIZED CUP AND POSTERIOR PLATING IN EARLY CRINOIDS (Site not responding. Last check: 2007-10-09)
		Thus, the CBPC has a perradial orientation in stem-group and dicyclic crinoids, but must have rotated 36 degrees to an interradial orientation in monocyclic crinoids.
		Correct plate homologies can be re-established in dicyclic crinoids by renaming infrabasals as basals and the previous overlying "basals" as epibasals, following Jaekel, 1918.
		Two derived posterior plate arrangements involving the C-ray plating arose from within the wider CD interray plating of stem-group crinoids, a pattern continued in later camerates where it originates at or above the radials.
	gsa.confex.com /gsa/2001AM/finalprogram/abstract_24305.htm (440 words)

	VEGA 0.5 Quick Reference Manual: Functions in EXMPLS10.M (Site not responding. Last check: 2007-10-09)
		Meta[x,y] denotes the group operation in the metacyclic groups.
		MetaP[x,y] denotes the group operation in the metacyclic groups.
		StripInverse[{i,j,s,p,q,k}] constructs the inverse in the infinite group with operation Strip.
	www.ijp.si /vega/htmldoc/USAGES/EXMPLS10.HTM (499 words)

	I P M - Papers
		On the orthogonal basis of the symmetry classes of tensors associated with the dicyclic group
		A necessary and sufficient condition for the existence of orthogonal basis of decomposable symmetrized tensors for the symmetry classes of tensors associated with the dicyclic group is given.
		In particular we apply these conditions to the generalized quaternion group, for which the dimensions of the symmetry classes of tensors are computed.
	www.ipm.ac.ir /IPM/publications/ViewPaperInfo.jsp?PTID=65&school=Mathematics (84 words)

	[ref] 71 The Character Table Library
		Note that the Brauer table and the corresponding ordinary table of a group determine the decomposition matrix of the group (or the decomposition matrices of its blocks).
		Note that class fusions stored on library tables are neither guaranteed to be consistent for any two subgroups of a group and their intersection, nor tested to be consistent with respect to composition of maps.
		for symmetric groups; the remaining arguments specialise then the desired member of the series (see Generic Character Tables for a list of available generic tables).
	www.math.niu.edu /help/math/gap4/ref/CHAP071.htm (5881 words)

	[No title] (Site not responding. Last check: 2007-10-09)
		Subsequent attempts to resolve the crinoid history were all based on the arrangement of thecal plates relative to the position of the arms.
		Traditionally, the thecal plates were decribed as either monocyclic or dicyclic with the first row containing the arms as the "orals", the row just below the orals as the "radials", the next row as the "basals", and for the dicyclic groups, the last row as the "infrabasals".
		This long established precedence of thecal plate organization has forced the assumption that each row of plates bearing the same name were in fact homologous.
	www.ucmp.berkeley.edu /echinodermata/crinsys.html (243 words)

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