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Topic: Conjugacy class

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	PlanetMath: conjugacy class formula
		The conjugacy classes of a group form a partition of its elements.
		In a finite group, this means that the order of the group is the sum of the number of elements of the distinct conjugacy classes.
		This is version 3 of conjugacy class formula, born on 2002-11-27, modified 2002-12-11.
	planetmath.org /encyclopedia/ConjugacyClassFormula.html (100 words)

	Conjugacy class - Wikipedia, the free encyclopedia
		In mathematics, especially group theory, the elements of any group may be partitioned into conjugacy classes; members of the same conjugacy class share many properties, and study of conjugacy classes of non-abelian groups reveals many important features of their structure.
		For any character, the value of the character is the same for all members of a conjugacy class.
		The orbits of this action are the conjugacy classes, and the stabilizer of a given element is the element's centralizer.
	en.wikipedia.org /wiki/Conjugacy_class (887 words)

	Conjugacy (Site not responding. Last check: 2007-10-21)
		Construct a set of representatives for the conjugacy classes of G. The classes are returned as a sequence of triples containing the element order, the class length and a representative element for the class.
		Al := "Random": Construct the conjugacy classes of elements for a matrix group G using an algorithm that searches for representatives of all conjugacy of G by examining a random selection of group elements and their powers.
		Given a group G for which the conjugacy classes are known and an element x of G, return the designated representative for the conjugacy class of G containing x.
	www.math.uiuc.edu /Software/magma/text266.html (739 words)

	Conjugacy (Site not responding. Last check: 2007-10-21)
		Conjugacy and Sylow Theorems - Gallian, Ch 24 # 4,5, Fraleigh,...
		DC MetaData for: On Conjugacy Classes of Closed Subgroups and Stabilizers of Bor...
		Metadaten für Asymptotic geometry and growth of conjugacy classes of nonpositive...
	www.scienceoxygen.com /math/267.html (198 words)

	Conjugacy Classes of Subgroups
		Representatives for the conjugacy classes of elementary abelian subgroups for the group G. The subgroups are returned as a sequence of records having the same format as Subgroups.
		The elements of the poset correspond to the conjugacy classes of subgroups.
		The number of elements of the conjugacy class of subgroups e that lie in a fixed representative of the conjugacy class of subgroups f.
	www.math.uiuc.edu /Software/magma/text174.html (2372 words)

	[No title]
		This example suggests that perhaps in general the conjugacy class structure produced by endomorphisms with kernels is some multiple of the conjugacy class structure given by an induced automorphism on some quotient of the original group.
		The conjugacy classes themselves are not the same (under $\Ad{(1\,5\,2)}$, the identity element is generalized-conjugate to other elements of the group), but there is some bijection of $G$ onto itself which carries $id$-conjugacy classes precisely onto $\Ad{(1\,5\,2)}$-conjugacy classes.
		We conclude that $\phi$-conjugacy classes in $G$ are unions of columns of the table; two columns are in the same $\phi$-conjugacy class if and only if their headings are in the same conjugacy class in $Z^\phi_1$.
	www.rose-hulman.edu /Users/faculty/sherman/REU96/pramod.txt (4827 words)

	Research (Site not responding. Last check: 2007-10-21)
		My Ph.D. thesis, (Coxeter groups, conjugacy classes and relative dominance, 2000) was largely concerned with the lengths of elements in conjugacy classes of Coxeter groups.
		A flat conjugacy class of W is a class whose elements all have the same length.
		In [8] we give a method of calculating the minimal and maximal length of involutions in a conjugacy class of a Coxeter Group, equipped with an (arbitrary) element of that class.
	www.ma.umist.ac.uk /sp/Research.htm (688 words)

	[ref] 68 Class Functions
		to the cyclotomics that is constant on conjugacy classes of
		So two class functions are equal if and only if their lists of values are equal, no matter whether they are class functions of the same character table, of the same group but w.r.t.
		Class functions are row vectors of cyclotomics, scalar multiplication of a class function with a cyclotomic yields a class function, and the sum and the difference of two class functions with the same underlying character table (see UnderlyingCharacterTable) are again class functions of this table.
	www.mathematik.uni-kassel.de /gap4/ref/CHAP068.htm (6026 words)

	Conjugacy
		Construct a set of representatives for the conjugacy classes of G. The classes are returned as a sequence of tuples containing the order of the elements in the class, the class length and a representative element for the class.
		The class map M: G -> {1,..., n} for the group G, where n is the number of conjugacy classes of G. ClassRepresentative(G, x) : GrpPC, GrpPCElt -> GrpPCElt
		The power map M associated with the conjugacy classes of G. M describes where the elements of the conjugacy classes of G move under powers.
	www.math.niu.edu /help/math/magmahelp/text338.html (413 words)

	Conjugacy
		The radical quotient G/R is computed and its classes computed and represented as elements of G. To extend to the next larger quotient, a group is computed from each class which acts on the transversal.
		To compute the classes of G/R, the fusion/random algorithm of Cannon and Souvignier [CS97] is used.
		The limitations of the algorithm are that R may be trivial, in which case nothing is done except to call a different algorithm, or one or more of the sections may be so large as to prohibit computing the action on the transversal.
	www.math.niu.edu /help/math/magmahelp/text258.html (1182 words)

	Low Index Subgroups
		Subgroup class 6 Index 14 Length 7 Subgroup generators :- { a, b^-1 * a * b * a * b^-1 * a * b * a * b^-1 * a * b * a * b^-1 * a * b, b * a * b * a * b^-1 }
		Subgroup class 7 Index 14 Length 14 Subgroup generators :- { a, b^-1 * a * b * a * b^-1 * a * b^-1 * a * b * a * b * a * b^-1 * a * b, b * a * b * a * b^-1 }
		Subgroup class 8 Index 14 Length 7 Subgroup generators :- { a, b * a * b * a * b^-1 * a * b^-1 * a * b * a * b * a * b^-1 * a * b^-1, b^-1 * a * b * a * b }
	www.math.ufl.edu /help/magma/text235.html (1972 words)

	[ref] 36.9 Conjugacy Classes
		A conjugacy class is an external orbit (ExternalOrbit) of group elements with the group acting by conjugation on it.
		It is guaranteed that the class of the identity is in the first position, the further arrangement depends on the method chosen (and might be different for equal but not identical groups).
		A rational class is an external set (IsExternalSet) of group elements with the group acting by conjugation on it, but not an external orbit.
	wwwmaths.anu.edu.au /research.programs/aat/GAP/www/Manual4/ref/C036S009.htm (413 words)

	[ref] 65 Character Tables
		This means mainly that the ordering of conjugacy classes used for the various attributes of the character table cannot be changed; see Sorted Character Tables for how to compute a character table with a different ordering of classes.
		All those lists stored in the table that are related to the orderering of conjugacy classes (such as sizes of centralizers and conjugacy classes, orders of representatives, power maps, and all class functions) refer to the ordering of this list.
		All rearrangements of classes and characters are stable, i.e., the relative positions of classes and characters that are not distinguished by any relevant property is not changed.
	www.math.colostate.edu /WWWextra/manuals/gap/CHAP065.htm (8653 words)

	Random generation of Linear Codes -- Random generation of linear codes
		The conjugacy classes of the operating group, which is a direct product of two groups, can be described as pairs of the conjugacy classes of the two factors.
		The number of matrices in the conjugacy class of the normal form in (*) is given by
		In order to minimize the amount of work before the algorithm actually starts to generate codes it is useful to start the generation at once after having computed the information on the first conjugacy class, and evaluate further conjugacy classes and their probabilities only if required.
	www.mathe2.uni-bayreuth.de /frib/html/code_dw/code_dw_2.html (1428 words)

	Ian's home page (Site not responding. Last check: 2007-10-21)
		He is writing a monograph (Train track expansions of measured foliations, available at his web site) on train tracks and the solution to the conjugacy problem in the mapping class group, which is already at 297 pages, even though he hasn't started describing his solution to the conjugacy problem.
		His thesis and early papers were on the topic of the conjugacy problem, and he has made some important contributions to understanding the combinatorial structure of the mapping class groups, namely that they have an automatic structure.
		One interesting thing mentioned by Lee at lunchtime is that the number of conjugacy classes of a matrix element in SL(2,Z) is the class number of the quadratic field generated over Q by the eigenvalues of the matrix.
	www.math.uic.edu /~agol/blog/030227.html (516 words)

	Ninth Homework Solutions (Site not responding. Last check: 2007-10-21)
		Now the size of a conjugacy class divides the order of the group.
		G is an element in a conjugacy class of size p
		Prove that the number of elements in the conjugacy class containing g is prime to p.
	www.math.vt.edu /people/linnell/5114/Ahw9 (410 words)

	[ref] 64 Tables of Marks
		If the group doesn't know its lattice of subgroups or its conjugacy classes of subgroups then the table of marks and the conjugacy classes of subgroups are computed at the same time by the cyclic extension method.
		In the latter case, the class of the derived subgroups could not be uniquely determined, and the position of the class of derived subgroups is an entry of
		The return value is either the list of class numbers of those subgroups that have the right size and contain the subgroup and all subgroups that clearly contain it as a normal subgroup, or the class number of the normalizer if it is uniquely determined by these conditions.
	www.math.colostate.edu /WWWextra/manuals/gap/CHAP064.htm (4606 words)

	Conjugacy
		Create the classes of G assuming that the first elements of the tuples in Q form a complete set of conjugacy class representatives and the corresponding integer is the size of the conjugacy class.
		The centraliser of the representative element stored for conjugacy class number i in group G. The group computed is stored with the class table for reference by future calls to this function.
		The default values for the random class algorithm are adequate for a large variety of groups.
	www.math.lsu.edu /magma/text275.htm (1671 words)

	Permutation Loops
		Exactly one of those is the null class, so on each iteration the probability of reaching null is 2/n!.
		By the way, two elements of S_n are in the same conjugacy class if and only if they have the same cyclic structure, so the number of conjugacy classes of S_n is p(n), the number of partitions of n.
		Also, the number of members of a given conjugacy class can be computed from the corresponding partition.
	www.mathpages.com /home/kmath031.htm (1269 words)

	Finite groups in MAPLE5
		For example, the conjugacy class of [[1,2]] in the symmetric group S3 is:
		The conjugacy class of [[2,4]] in the symmetric group D4 is:
		Thus, for example, the 2nd column corresponds to the value of the character on the conjugacy class represented by (1,2), since it is of type [1,1,2].
	web.usna.navy.mil /~wdj/symm_gp.html (889 words)

	Conjugacy Classes of Elements
		, testing conjugacy is performed by transforming each element into a standard representative of its conjugacy class by an orbit-stabilizer process that works down a sequence of increasing quotients of G. Conjugacy testing for a group G in category
		: Construct the conjugacy classes of elements for a permutation or matrix group G using an algorithm that searches for representatives of all conjugacy classes of G by examining a random selection of group elements and their powers.
		A representation of G/R is computed using an algorithm of Derek Holt and its classes computed and represented as elements of G. To extend to the next larger quotient, a group is computed from each class which acts on the transversal.
	www.math.lsu.edu /magma/text261.htm (1080 words)

	AMCA: Conjugacy classes of the group of units in group algebras of a finite $p$-groups by Adalbert Bovdi (Site not responding. Last check: 2007-10-21)
		AMCA: Conjugacy classes of the group of units in group algebras of a finite $p$-groups by Adalbert Bovdi
		Conjugacy classes of the group of units in group algebras of a finite p-groups
		The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Mathematical Conference Abstracts.
	at.yorku.ca /c/a/f/e/21.htm (298 words)

	Conjugacy
		: Construct the conjugacy classes of elements for a matrix group G using an algorithm that searches for representatives of all conjugacy of G by examining a random selection of group elements and their powers.
		Before describing the effect of these parameters, some definitions are needed: A mapping f: G -> I is called a class invariant if f(g) = f(g^h) for all g, h in G. In matrix groups, the primary invariant factors are used where possible, or the characteristic or minimal polynomials otherwise.
		Given a group G, and a sequence Q of k distinct elements of G, one from each conjugacy class, use Q to define the classes attribute of G. The sequence Q may be either a sequence of elements of G or, preferably, a sequence of pairs
	www.math.lsu.edu /magma/text298.htm (1164 words)

	ABSTRACTS OF PAPERS ERICH W. ELLERS
		Conjugacy classes of involutions in the Lorentz group Omega(V) and in SO(V).
		Let G be a simple and simply-connected algebraic group that is defined and quasi-split over a field K. We investigate properties of intersections of Bruhat cells of G with conjugacy classes C of G, in particular, we consider the question, when is such an intersection not empty.
		We give some description of the intersections of noncentral conjugacy classes of G with certain Gauss cells, which we call Coxeter cells.
	www.math.toronto.edu /ellers/abstracts.html (817 words)

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