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	Order theory - Wikipedia, the free encyclopedia
		Order theory is a branch of mathematics that studies various kinds of binary relations that capture the intuitive notion of a mathematical ordering.
		Orders appear everywhere - at least as far as mathematics and related areas, such as computer science, are concerned.
		These are graphs where the vertices are the elements of the poset and the ordering relation is indicated by both the edges and the relative positioning of the vertices.
	en.wikipedia.org /wiki/Order_theory (4023 words)

	Cyclic group - Wikipedia, the free encyclopedia
		Every cyclic group is isomorphic to the group { 0, 1, 2,..., n − 1 } under addition modulo n, or Z, the additive group of all of integers.
		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.
		The cycle graphs of finite cyclic groups are all n-sided polygons with the elements at the vertices.
	en.wikipedia.org /wiki/Cyclic_group (1302 words)

	PlanetMath: cyclic group
		Note that the isomorphisms mentioned in the previous paragraph imply that all cyclic groups of the same order are isomorphic to one another.
		It follows from Lagrange's Theorem that every group of prime order is cyclic.
		This is version 15 of cyclic group, born on 2002-02-19, modified 2006-04-27.
	planetmath.org /encyclopedia/CyclicSubgroup.html (253 words)

	Info Node: (elisp)Cyclic Window Ordering
		Cyclic Ordering of Windows ========================== When you use the command `C-x o' (`other-window') to select the next window, it moves through all the windows on the screen in a specific cyclic order.
		If the first split was vertical (into windows one above each other), and then the subwindows were split horizontally, then the ordering is left to right in the top of the frame, and then left to right in the next lower part of the frame, and so on.
		If the first split was horizontal, the ordering is top to bottom in the left part, and so on.
	www.cs.cmu.edu /cgi-bin/info2www?(elisp)Cyclic%20Window%20Ordering (692 words)

	The Order of Time - Politics : welcome
		Like we right away stated in the preface is each form of order in the universe in fact cyclic because the strictly linear for itself, and certainly from an organic perspective, is only destructive, like the shrapnel is and all the rest that in a linear fashion spreads after the exploding of a bomb.
		So it is, as far as it concerns the cyclic order of time, the nature on this planet and not anything else, and they who see it differently are not really of this planet, are scientifically spoken not really 'with their feet on the ground'.
		It was written before the writer himself was really in order with the time of the ethereal nature, but, because of that in deep thought about it, with a well-filled bookcase had made a begin with the filognostic integration in the form of meditations on the righteousness of action and the uniting of the spirit.
	www.theorderoftime.com /bar/sciencew.html (2318 words)

	PlanetMath: a finite ring is cyclic if and only its order and characteristic are equal
		A finite ring is cyclic if and only if its order and characteristic are equal.
		"a finite ring is cyclic if and only its order and characteristic are equal" is owned by mathcam.
		This is version 5 of a finite ring is cyclic if and only its order and characteristic are equal, born on 2003-03-11, modified 2006-07-07.
	planetmath.org /encyclopedia/Characteristic2.html (144 words)

	XEmacs User's Manual - Other Window
		When there are more than two windows, the command moves through all the windows in a cyclic order, generally top to bottom and left to right.
		From the rightmost and bottommost window, it goes back to the one at the upper left corner.
		A negative numeric argument moves around the cycle in the opposite order.
	www.tau.ac.il /cc/pages/docs/xemacs/xemacs_133.html (239 words)

	Cyclic order - Wikipedia, the free encyclopedia
		In combinatorial mathematics, a cyclic order on a set X with n elements is an arrangement of X as on a clock face, for an n-hour clock.
		An infinite set can also be ordered cyclically.
		The precise definition is more involved, but the idea is the same: arrange the (infinitely many) members of the set around a circle.
	en.wikipedia.org /wiki/Cyclic_order (295 words)

	Numbers game - 25 January 1997 - New Scientist
		Cyclic numbers occur when we have a prime number, say p, and its reciprocal 1/p in decimal form is recurring with (p-1) digits.
		A cyclic number is an integer of n digits with the following property: when multiplied by any number from 1 through n, the product contains the same n digits in the same cyclic order.
		All cyclic numbers can be multiplied by the decimal representation of 1/p where p is a prime, however not all primes work.
	www.newscientist.com /backpage.ns?id=lw314 (1163 words)

	Cyclic Groups and Subgroups
		Thus for a cyclic group we have the definition that all the elements may be generated from a single element together with its inverse.
		The order of the subgroup must be a divisor of the order of G (Lagrange's Theorem) but since G is of prime p order the only divisors of p are 1 and p itself.
		Thus the order of a cyclic subgroup and the order of its generator are the same number.
	members.tripod.com /~dogschool/cyclic.html (2281 words)

	Playing Cards
		To construct a constant order card flexagon, the outer faces in the map are numbered in the order in which they are approached by the traversal of the hinge network (large numbers in fig.
		Since the flexagon plan is cyclic in structure, we have no way of knowing the order of the two pats, unless we invent some way of distinguishing the two different hinges joining the two pats.
		To keep in mind that it is a break in the cyclic constant order, the two leaves connected by the unfolded-together hinge can be encircled: the lowest term of the left-hand pat, and the highest term of the right-hand pat.
	delta.cs.cinvestav.mx /~mcintosh/comun/flexagon/node28.html (1567 words)

	Math 417 Practice Problems
		Cyclic groups are abelian, so we are done if G is cyclic.
		Thus there is one more element of G of order p, say x, and it cannot be a power of g.
		Since the order, which is 5, of b is the least common multiple of the lengths of these cycles, we know that every cycle in the product has length 1 or 5, and at least one of them has length 5.
	www.math.unl.edu /~bharbourne1/M417Spr03/PracticeProbs.html (864 words)

	THE ORDER OF TIME - INFORMATION: ABOUT THE ORDER OF TIME more.
		The existence of a causal order thus is unmistakably true: first one makes love and next one sees a child born, first one sees lightning and then one hears the thunder, that suffers no doubt.
		Causality thus is evidently linked, if not equal to, a cyclic order in which things give rise to each other: hydrogen and oxygen combined give water, and so, from one chemical reaction to the other, is a complexity generated in which we next find ourselves back as created, or conditioned, beings depending on circumstances.
		Next to the pages on the order of the time of the moon, de sun and the galaxy is added is a basic concept of philosophy putting all this science of time in a meaningful perspective.
	www.theorderoftime.com /info/i/toot.html (2290 words)

	Cyclic subgroups, Cauchy's Theorem.
		Another way to state this is to say that the order of the element a is the smallest positive integer n such that
		The order of any element of a group will divide the order of the group.
		The existence of such cyclic subgroup, for some orders, is true as well.
	hemsidor.torget.se /users/m/mauritz/math/alg/cyclsub.htm (320 words)

	Cyclic Window Ordering (Site not responding. Last check: 2007-11-02)
		Cycling Ordering of Windows =========================== When you use the command `C-x o' (`other-window') to select the next window, it moves through all the windows on the screen in a specific cyclic order.
		But it may go down first or go right first, depending on the order in which the screen, or the windows within the screen, were split.
		If the screen was first split horizontally, the ordering is top to bottom in the left part, and so on.
	www.cis.ksu.edu /VirtualHelp/Info/gnu/elisp.Cyclic_Window_Ordering.html (400 words)

	TH info from binary representation of the move No.
		Odd disks are always moved according to the cyclic order, even disks in the opposite order.
		In move 1 (k=1), disk 1 is moved from peg 0 to peg 2, in move 3 (k=3) from peg 2 to peg 1, in move 5 (k=5) from peg 1 to peg 0, etc., always in the cyclic order.
		If the disk moved in step 2 is also odd, it must be different from disk 1, and so be moved from peg that follows in the cyclic order the final peg of the move of disk 1 of step 1.
	hanoitower.mkolar.org /moves.html (772 words)

	[No title] (Site not responding. Last check: 2007-11-02)
		A generalization to cyclic $n$-tuples --- an $n$-tuple $(x_1,x_2,\ldots, x_n)$ for which $\langle x_1,x_2,\ldots,x_n \rangle$ is cyclic --- is also established.
		Since $q^k$ is the maximum order of an element of $\Z_{q^k} \oplus Z_{q^k}$, it follows that $(c,d) \in \langle (a,b)\rangle$.
		Thus $\langle B \rangle$ is a cyclic subgroup of $\Z_{q^{m+1}}$ containing no elements of order $q^{m+1}$.
	www.rose-hulman.edu /Users/reu/reu92/reu92jk/testscreen (1042 words)

	Isomorphism
		They are all of order 4 (but that's not what makes them isomorphic) and are cyclic groups.
		For example, there are 120 bijections between two groups of order 5 (and 24 of these map the identity to the identity).
		For example, the cyclic group of order 4 has two elements of order 4 whereas the Klein 4 group has no elements of order 4.
	www.math.csusb.edu /notes/advanced/algebra/gp/node19.html (393 words)

	GNU Emacs Lisp Reference Manual: Cyclic Window Ordering
		But it may go down first or go right first, depending on the order in which the windows were split.
		If the first split was vertical (into windows one above each other), and then the subwindows were split horizontally, then the ordering is left to right in the top of the frame, and then left to right in the next lower part of the frame, and so on.
		If the first split was horizontal, the ordering is top to bottom in the left part, and so on.
	www.ualberta.ca /~neitsch/sunsite.ualberta.ca/Mirror/gnu/Manuals/elisp-manual-21-2.8/html_node/elisp_431.html (440 words)

	round-robin on bind9.2.1 (Site not responding. Last check: 2007-11-02)
		RRset Ordering When multiple records are returned in an answer it may be useful to configur e the order of the records placed into the response.
		The legal values for ordering are: fixed Records are returned in the order they are defined in the zone file.
		RRset Ordering > When multiple records are returned in an answer it may be=20 > useful to configure > the order of the records placed into the response.
	www.webservertalk.com /message927184.html (1754 words)

	p-cyclic over Other Characteristics
		The only group of order p is cyclic, hence f/k is a cyclic extension of order p.
		Thus the order of d is divisible by p.
		Thus the order of d is divisible by p-1.
	www.mathreference.com /fld-slv,npcyc.html (696 words)

	Homework 2
		Any subgroup of a cyclic group must have order dividing the order of the cyclic group, and (a) and (b) are cyclic, so the order of (a) intersect (b) must divide both m and n.
		The number of subgroups of a cyclic group is the number of divisors of the order of that group.
		G must have order which is a multiple of 10 and 25, thus a multiple of 50.
	www.math.umn.edu /~stanton/5245/oldprob5245.html (3458 words)

	Cyclic Groups (Site not responding. Last check: 2007-11-02)
		We have shown that any two infinite cyclic groups are isomorphic.
		Thus in the case of cyclic groups of finite order, the converse of Lagrange's theorem is true, viz, for each
		We shall see presently that (e) really characterizes finite cyclic groups, but first we establish an extremely useful theorem, which will be used many times.
	web.usna.navy.mil /~wdj/tonybook/gpthry/node27.html (623 words)

	[No title]
		Generators of cyclic groups Properties: any cyclic group is isomorphic to Z or Zn.
		of Cyclic groups: (i) being “cyclic” is “hereditary”; (ii) the order is a “complete invariant” (on subgroups of cyclic groups) The Subgroup Lattice Def.; Examples; Number of elements of each order: - Interpretation: generators of subgroups; - Euler’s phi function ((d); Prop.U(d)=((d).
		invertible as a function - Isomorphisms preserve: e, inverse, order, Abelian, cyclic (with proofs); - Inner automorphisms Inn(G), and Aut(G) are groups (Inn(G) is a subgroup of Aut(G)) (proofs) - Left regular representation of a group; Cayley Th.
	www.ilstu.edu /~lmiones/336rvs05.DOC (846 words)

	Citations: A Point Set Approach in Geometric Modelling - Gursoz, Prinz (ResearchIndex)
		They are stored in a cyclic order so that different zones around an....
		A disk is a cyclic list of faces that separate a zone of the neighbourhood of the vertex from another zone.
		On each side of a cone, there is a disk which is the cyclic list of the faces that form the cone.
	citeseer.ist.psu.edu /context/1118662/0 (289 words)

	Groups
		If every element of a group can be generated by a single element, the group is said to be cyclic.
		is not cyclic: there is no element which generates the whole group.
		The order of an element should not be confused with the order of a group, which is the number of elements in the group.
	www.engineering.usu.edu /classes/ece/7670/lecture3/node1.html (557 words)

	XEmacs Lisp Reference Manual - Cyclic Window Ordering
		) to select the next window, it moves through all the windows on the screen in a specific cyclic order.
		determines whether the minibuffer is included in the window order.
		, then the cyclic ordering includes the minibuffer window even if it is not active.
	www.tau.ac.il /cc/pages/docs/xemacs/lispref_426.html (448 words)

	ABSTRACT ALGEBRA: OnLine Study Guide, §3.5 Solved Problems
		is cyclic since it has an element whose order is equal to the order of the group.
		Solution: If G is cyclic of order 2n, for some positive integer n, then it follows from Theorem 3.5.2 that G is isomorphic to Z
		In any isomorphism, cyclic subgroups would correspond to cyclic subgroups, and so it is impossible for this group to be isomorphic to the quaternion group, which has 3 cyclic subgroups of order 4.
	www.math.niu.edu /~beachy/abstract_algebra/study_guide/soln35.html (628 words)

	Polynomial order (Site not responding. Last check: 2007-11-02)
		The order of P is the smallest positive integer n such that P(x) divides x
		If you enter a reducible polynomial, the orders of all its non-linear factors will be computed and presented.
		In order to test the browser you are using, please type the word
	wims.unice.fr /wims/en_tool~algebra~polyorder.en.html (202 words)

	Klein and Cyclic group
		The fact the group is of order 4 is significant...
		There are only two groups of order two ("up to isomorphism"), the cylic group of order 4 and the Klein group.
		the integers mod 6 under addtion are cyclic of order 6.
	www.physicsforums.com /showthread.php?t=12483 (1095 words)

	Info Node: (emacs)Other Window (Site not responding. Last check: 2007-11-02)
		After the rightmost and bottommost window, it goes back to the one at the upper left corner.
		A numeric argument means to move several steps in the cyclic order of windows.
		A negative argument moves around the cycle in the opposite order.
	www.cs.cmu.edu /cgi-bin/info2www?(emacs)Other%20Window (278 words)

	Math 720
		Let G be cyclic of order n and let k be (another, but not necessarily different) integer.
		If k divides n, then show that there is a unique subgroup of G of order k.
		If k does not divide n, then show there is no subgroup of G of order k.
	www.ndsu.nodak.edu /ndsu/coykenda/M720.1.F2000.htm (161 words)

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