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Topic: Alternating group

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Trivial group

Janko group

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	PlanetMath: alternating group is a normal subgroup of the symmetric group
		alternating group is a normal subgroup of the symmetric group
		"alternating group is a normal subgroup of the symmetric group" is owned by mathcam.
		This is version 2 of alternating group is a normal subgroup of the symmetric group, born on 2003-06-23, modified 2004-04-30.
	planetmath.org /encyclopedia/AlternatingGroupIsANormalSubgroupOfTheSymmetricGroup.html (108 words)

	Alternating group
		In mathematics an alternating group is the group of even permutations of a set.
		The alternating group on the set {1,...,n} is called the alternating group of degree n, or the alternating group on n letters and denoted by A
		For instance: {1234, 1342, 1423, 2143, 2314, 2431, 3124, 3241, 3412, 4132, 4213, 4321} is the alternating group of degree 4.
	www.ebroadcast.com.au /lookup/encyclopedia/al/Alternating_group.html (118 words)

	PlanetMath: examples of finite simple groups
		The simplicity of the alternating groups is an important fact that Évariste Galois required in order to prove the insolubility by radicals of the general polynomial of degree higher than four.
		Cross-references: Janko groups, regular graphs, automorphisms, sequences, infinite, projective special linear groups, commutators, centers, sources, polynomial, simplicity of the alternating groups, argument, simple, index, transposition, between, bijection, odd, permutation, odd permutations, homomorphism, kernel, normal subgroup, symmetric group, even permutations, abelian, Cauchy's theorem, order, cyclic groups, isomorphism, simple groups, finite, groups
		This is version 12 of examples of finite simple groups, born on 2002-11-04, modified 2004-11-17.
	planetmath.org /encyclopedia/ExamplesOfFiniteSimpleGroups.html (360 words)

	Characters of the Alternating Group (Site not responding. Last check: 2007-10-20)
		So the irreducible characters of the alternating group may be indexed by partitions in the same way as those of the symmetric group.
		Return the value of the character of the alternating group of degree n indexed by the partition pa of weight n on the permutation pe.
		Return the value of the ith character of the alternating group of degree n indexed by the self conjugate partition pa of weight n on the permutation pe.
	magma.maths.usyd.edu.au /magma/htmlhelp/text946.htm (321 words)

	Alternating group A5 (Site not responding. Last check: 2007-10-20)
		Remark: This is the Coxeter group (2, 3, 5).
		Remark: This is the Coxeter group (2, 4, 5; 3).
		Remark: This is the Coxeter group (2, 5, 6; 2).
	web.mat.bham.ac.uk /J.N.Bray/Pres/A5.html (85 words)

	[No title] (Site not responding. Last check: 2007-10-20)
		The permutation group generated by (1,2) and (2,3) is equal to the symmetric group on {1,2,3}.
		The permutation group generated by (1,2,3) and (3,4,5) is equal to the alternating group on {1,2,3,4,5}.
		The group is generated by the permutations (1,2,...,n) and (1,n)(2,n-1)(3,n-3)....(n/2-1/2,n/2+1/2) if n is odd and by the permutations (1,2,...,n) and (1,n)(2,n-1)(3,n-3)....(n/2-1,n/2+1) if n is odd.
	www.win.tue.nl /~amc/oz/om/cds/permgp2.html (339 words)

	Alternating group - Wikipedia, the free encyclopedia
		It is the kernel of the signature group homomorphism sgn : S
		As in the symmetric group, the conjugacy classes in A
		There are some isomorphisms between some of the small alternating groups and small groups of Lie type.
	en.wikipedia.org /wiki/Alternating_group (237 words)

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		The number of trials taken to reach the criterion of extinction was greatest for the 50% random group and least for the 50% alternating group, with the 80% random group intermediate between the two.
		The results of the study indicates that the number of trials taken to reach the criterion of extinction was greatest for the 50% random group and least for the 50% alternating group, with the 80% random group intermediate between the two.
		The mean number of trials to extinction for the three groups were 10.63 trials for the 50% random group, 9.21 trials for the 80% random group, and 5.41 trials for the 50% single-alternation group.
	www.ecsu.ctstateu.edu /personal/faculty/nebelkopf/DUMP/dump/PREVpages/Halloween98/MAINPAGE/experimental/DAWNJUR.htm (1719 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)

	Learn more about List of small groups in the online encyclopedia. (Site not responding. Last check: 2007-10-20)
		For each order, all groups of that order up to group isomorphism are listed.
		: the symmetric group of degree n, containing the n permutations of n elements.
		The group theoretical computer algebra system GAP (available for free at http://www.gap-system.org/) contains the "Small Groups library": it provides access to descriptions of the groups of "small" order.
	www.onlineencyclopedia.org /l/li/list_of_small_groups.html (306 words)

	Historia Matematica Mailing List Archive: [HM] Alternating group (Site not responding. Last check: 2007-10-20)
		saw that Jordan's proof of the simplicity of the alternating group of degree
		He draws from this theorem the conclusion that the alternating group on n
		group on n letters shows which is sharply (n - 2)-transitive.
	sunsite.utk.edu /math_archives/.http/hypermail/historia/jul99/0051.html (395 words)

	Alternation of Antiretroviral Drug Regimens for HIV Infection: Efficacy, Safety and Tolerability at Week 96 of the ...
		Virological suppression was seen in 46%, 48% and 58% of patients in groups A, B and C, respectively, in the intention-to-treat analysis and in 75%, 76% and 97% in the on-treatment analysis (A vs C: P=0.014; B vs C: P=0.016; A vs B: P=0.849).
		Alternating therapy was associated with a similar impact on CD4+ counts in comparison with the standards of care regimens, as well as a lower mitochondrial DNA/nuclear DNA (mtDNA/nDNA) ratio decrease in the mitochondrial substudy performed on 37 patients.
		The frequency and intensity of adverse events in the alternating group decreased during subsequent cycles.
	www.hivandhepatitis.com /recent/toxicities/mitochondrial/012605_b.html (358 words)

	Representations Of The Alternating Group Which Are Irreducible Over Subgroups - Kleshchev, Sheth (ResearchIndex) (Site not responding. Last check: 2007-10-20)
		0.4: Branching of Modular Representations of the Alternating Groups - Bessenrodt, Olsson (1998)
		Kleshchev and J. Sheth, Representations of the alternating group which are irreducible over subgroups, preprint, University of Oregon, 1999.
		4 Branching of modular representations of the alternating grou..
	citeseer.ist.psu.edu /kleshchev99representations.html (697 words)

	PlanetMath: simplicity of the alternating groups (Site not responding. Last check: 2007-10-20)
		"simplicity of the alternating groups" is owned by rmilson.
		This is version 9 of simplicity of the alternating groups, born on 2002-11-04, modified 2004-02-25.
		(Group theory and generalizations :: Abstract finite groups :: Simple groups: alternating groups and groups of Lie type)
	planetmath.org /encyclopedia/SimplicityOfA_n.html (316 words)

	Identification
		The first functions described in this subsection detect whether or not a permutation group is alternating or symmetric in its natural representation.
		The group G should be known to be isomorphic to the symmetric group S_n for some n >= 8.
		The group G should be known to be isomorphic to the alternating group A_n for some n >= 9.
	www.math.lsu.edu /magma/text285.htm (846 words)

	Determining the Galois Group of a Polynomial (Site not responding. Last check: 2007-10-20)
		With the exception of primes that divide the discriminant (which for this equation is (-2^16)(7^4)), the polynomial will have no repeated roots, so if there are 6 distinct values of x between 0 and p-1 such that f(x)=0 (mod p), then f splits in F_p[x].
		Now, by a theorem of Cebotarev we know the proportion of all primes relative to which f(x) splits is asymptotic to 1/order(G) where G is the galois group of f.
		Clearly the order of the group is not 720 (which it would have to be if the Galois group was S_6), so it must be a proper subgroup.
	www.mathpages.com /home/kmath485.htm (326 words)

	[No title] (Site not responding. Last check: 2007-10-20)
		A spiral arrangement of leaves on a stem, with one leaf at a node.
		] A navigation light having groups of total eclipses at regular intervals and having color variations, the duration of light being equal to or greater than that of darkness.
		Any of a group of double sulfates of trivalent metals such as aluminum, chromium, or iron and a univalent metal such as potassium or sodium.
	www.accessscience.com /Dictionary/A/A22/DictA22.html (2206 words)

	[No title]
		Finite simple groups and localization Jose L. Rodriguez, Jer^ome Scherer and Jacques Thevenaz * Abstract The purpose of this paper is to explore the concept of localization, wh* ich comes from homotopy theory, in the context of finite simple groups*.
		We prove that all sporadic * simple groups* (except possibly the Monster) and several groups of Lie type are co* nnected to alternating* groups.
		The Conway groups are com* plete, the smaller ones are maximal simple subgroups of Co1 and there is a unique conj *ugacy class of each of them in Co1 as indicated in the ATLAS [4, p.180].
	hopf.math.purdue.edu /Rodriguez-Scherer-Thevenaz/simplegroups.txt (6910 words)

	Senior Thesis Pages: Melissa Banister (Site not responding. Last check: 2007-10-20)
		We present the results of an investigation into the representation theory of the alternating and dihedral groups and explore how their irreducible representations can be distinguished with the use of class sums.
		The four class sums corresponding to the 2-cycles, 3-cycles, 4-cycles, and 5-cycles are sufficient to distinguish all of the irreducible representations of the symmetric group of order 41.
		A natural question to ask is whether the irreducible representations of other finite groups can be distinguished with a small number of class sums relative to the order of the group.
	www.math.hmc.edu /seniortheses/archives/2004/mbanister (245 words)

	Cancer Spectrum: Medline Abstract (Site not responding. Last check: 2007-10-20)
		Alternating mitomycin C and bacillus Calmette-Guerin instillation prophylaxis for recurrent papillary (stages Ta to T1) superficial bladder cancer.
		Recurrence rates during the instillation period were 1.01 in group 1 and 0.86 in group 2 (p = 0.376).
		CONCLUSIONS: Efficacy of alternating mitomycin C and BCG was equal to mitomycin C monotherapy, and both methods were effective in prophylaxis of recurrent papillary bladder cancer.
	jncicancerspectrum.oxfordjournals.org /cgi/medline/pmid;8648837 (316 words)

	ASHM - Publications
		Weight increase was greater in the simultaneous group (+5.8 kg) than in the alternating group (+1.9 kg).
		There were 4 opportunistic infections in the simultaneous group and 10 in the alternating group, but the time to the first OI did not differ between the groups (p=0.16).
		One patient in the simultaneous group died of fulminant pancreatitis.
	www.ashm.org.au /index.php?&PageCode=954 (891 words)

	[No title] (Site not responding. Last check: 2007-10-20)
		In permutation groups of large degree, the number of Schreier generators to be processed becomes too large, and the deterministic Schreier-Sims algorithm becomes impractical.
		This construction is complemented by a verify-routine that either proves the correctness of the stabilizer chain or causes the extension of the chain to a correct one.
		The problem is that the group has a stabilizer subgroup $G^{(i)}$ such that the fundamental orbit $O^{(i)}$ is split into a lot of orbits when we stabilize $b_i$ and one additional point of $O^{(i)}$.
	www-groups.dcs.st-and.ac.uk /~gap/Manuals/doc/build/grpperm.msk (3012 words)

	A randomized phase II study of alternating and sequential regimens of docetaxel and doxorubicin as first-line ...
		A randomized phase II study of alternating and sequential regimens of docetaxel and doxorubicin as first-line chemotherapy for metastatic breast cancer -- Paridaens et al.
		alternating and sequential regimens of docetaxel and doxorubicin
		with the alternating regimen; 87% with the sequential regimen).
	annonc.oxfordjournals.org /cgi/content/full/14/3/433 (4285 words)

	[ref] 41 Permutation Groups
		A permutation group is a group of permutations on a finite set Ω; of positive integers.
		A group is a natural alternating group if it is a permutation group acting as alternating group on its moved points.
		For groups that are known to be natural symmetric or natural alternating groups, very efficient methods for computing membership, conjugacy classes, Sylow subgroups etc. are used.
	www.gap-system.org /Manuals/doc/htm/ref/CHAP041.htm (4261 words)

	Alternating groups.
		This means that all permutations could be revritten as alternations, and that all these altenations could be done against one element.
		The first alternating group that is not isomorphic to any modulo n group, Z
		This is interesting enough for an extra examination of that group, and this is the
	hemsidor.torget.se /users/m/mauritz/math/alg/alt.htm (274 words)

	The alternating group
		The following remarkable result about the alternating group will not be needed to understand the structure of the Rubik's cube.
		However, the theorem below is interesting because of its connection with the fact (due to N. Abel and E. Galois) that you cannot solve the general polynomial of degree 5 or higher using radicals, i.e., that there is no analog of the quadratic formula for polynomials of degree 5 or higher.
		It turns out that the permuation which puts the edges in the correct position must be an even permutation.
	web.usna.navy.mil /~wdj/book/node189.html (283 words)

	Symmetric and Alternating Groups (Site not responding. Last check: 2007-10-20)
		, is the group of all possible permutations on n letters.
		, is the group of all even permutations on n letters.
		This can be seen by placing any vertex in position, then the next, then the next, and so on, and reflecting if the last two must be swapped.
	www.mathreference.com /grp,sym.html (122 words)

	untitled
		The independent variable for the study is the discriminability of the reinforcement pattern, and will be operationally defined by the reinforcement pattern administered to the three groups (50% single-alternation, 80% random reinforcement, and 50% random reinforcement schedules).
		Specifically, the 50% single-alternation group should show significantly faster extinction than the 80% random reinforcement group, with the 50% random reinforcement group taking the longest to extinguish the learned response.
		group would recognize the extinction shift earlier than the 50% random group.
	www.ecsu.ctstateu.edu /personal/faculty/nebelkopf/methods/dawnjur.htm (1557 words)

	PSL(2,7) (Site not responding. Last check: 2007-10-20)
		The projective special linear group G = PSL(2,7) is a finite group in mathematics that has important applications in algebra, geometry, and number theory.
		It is the automorphism group of the Klein quartic, and it is the second-smallest nonabelian simple group, next to the alternating group A
		Then G = PSL(2,7) is defined to be the quotient group SL(2,7) / obtained by identifying I and −I. In this article, we let G denote any group isomorphic to PSL(2,7).
	www.worldhistory.com /wiki/P/PSL(2,7).htm (673 words)

	Quotient groups
		Galois's work was the first to begin to illuminate the ``basic building blocks'' of the collection of finite groups (i.e., the analog of the idea that atoms are the ``basic building blocks'' of molecules).
		Roughly speaking, one may say that the basic building blocks of finite groups are those the groups which have no proper non-trivial normal subgroups.
		Intuitively, this is because a non-trivial quotient group (by a normal subgroup) is closely related to the original group but smaller in size (and hence perhaps subject to analysis by an inductive argument of some type).
	web.usna.navy.mil /~wdj/book/node190.html (328 words)

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