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	Automorphism - Wikipedia, the free encyclopedia
		In graph theory an automorphism of a graph is a permutation of the nodes that preserves edges and non-edges.
		An automorphism of a differentiable manifold M is a diffeomorphism from M to itself.
		In Riemannian geometry an automorphism is a self-isometry.
	en.wikipedia.org /wiki/Automorphism (875 words)

	[No title]
		In the former category, an automorphism fixes the specified subfield R, while in the latter category an automorphism permutes an infinite number of isomorphic copies of R. Thus, in the sense of category theory, the "field C" is not isomorphic to the "field C with a specified subfield R".
		There are plenty of order 2 automorphisms of C. Let t be a non-real transcendental number, and take K to be a maximal real-closed subfield of C containing Q(t) (use Zorn's Lemma).
		The automorphism group of K is elementary abelian of order 4, so any automorphism has order 1 or 2, and its fixed field on K is either K itself, or a quadratic field.
	www.hut.fi /~ppuska/mirror/Lounesto/RCHO (1101 words)

	Automorphism (Site not responding. Last check: 2007-10-07)
		In graph theory an automorphism of a graph is a permutation of the nodes that maps the graph to itself.
		The former corresponding to automorphisms coming from "conjugation" by elements of the object itself, and the latter being everything else.
		An inner automorphism is then an automorphism corresponding to conjugation by some element a.
	www.yotor.com /wiki/en/au/Automorphism.htm (875 words)

	NTU Info Centre: Order isomorphism (Site not responding. Last check: 2007-10-07)
		In the mathematical field of order theory an order isomorphism is a special kind of monotone function that constitutes a suitable notion of isomorphism for partially ordered sets.
		Whenever two partially ordered sets are order isomorphic, they can be considered to be "essentially the same" in the sense that one of the orders can be obtained from the other just by renaming of elements.
		An order isomorphism from (S, ≤) to itself is called an order automorphism.
	www.nowtryus.com /article:Order_isomorphism (200 words)

	[No title]
		Let G be the dihedral group of order 8 (the group of symmetries of the square).
		It has an elementary-abelian subgroup H of order 4 generated by the reflections across the edges of the square.
		Then there is an automorphism s of H which exchanges b and c, but it can't be lifted to G because c is in the center of G, while b is not.
	www.math.niu.edu /~rusin/known-math/95/extend.auto (836 words)

	ATLAS: Linear group L3(4)
		An automorphism of order 3 can be obtained by mapping (a,b) to (a, (abb)^-2babababbab(abb)^4).
		An automorphism of order 6 can be obtained by mapping (a,b) to ((ab)^-2bab, (abb)^-2babababbab(abb)^4).
		An automorphism of order 3 can be obtained by mapping (c,d) to (d^-1cd, (abb)^-3b(abb)^3).
	web.mat.bham.ac.uk /atlas/html/L34.html (1033 words)

	Steiner 2-designs
		There are two S(2,4,37) designs with automorphisms of order 37 and 284 with automorphisms of order 11.
		In a paper under preparation automorphisms of order 2 and 3 are studied and used to find many more examples.
		There are exactly three S(2,5,45) designs with automorphisms of order 5 (full automorphism groups are of order 360, 160 and 40).
	www.math.hr /~krcko/results/steiner.html (447 words)

	Groups of small order.
		All groups of prime order p are isomorphic to C_p, the cyclic group of order p.
		There are the three of order 2 generated by (1 2), (1 3) and (2 3), and the one of order 3 generated by (1 2 3).
		This is the same subgroup lattice structure as for the lattice of subgroups of C_8 x C_2, although the groups are of course nonisomorphic.
	www.math.usf.edu /~eclark/algctlg/small_groups.html (1543 words)

	[No title] (Site not responding. Last check: 2007-10-07)
		The elements of order 3 are of two types, those that square the 7-elements and those that 4th power them.
		It has a cyclic derived factor group and an element of order 3 is not inverted by any automorphism.
		We can let N be the centralizer of C in A, where C is the subgroup of order 7 in H. Then x maps to an element of order 3 in A/N and since A/N is abelian, this element is not conjugate to its inverse.
	www.bath.ac.uk /~masgcs/problem/commentary2.html (208 words)

	Action of automorphisms, Order(G) = 64, Hall-Senior number = 15
		The group of outer automorphisms of G has order
		Order of the class of the automorphism in the outer automorphism group:
		This automorphism induces the identity homomorphism on cohomology
	www.math.uga.edu /~lvalero/cohohtml/groups_64_15_auto.htm (183 words)

	ABSTRACTS DROSTE
		For both finite and infinite chain cases the simple automorphism groups split into two classes: those where there is a bound (<12) on the number of conjugates required to express one non-identity element in terms of another, and those in which there is no such bound.
		Using combinatorial methods, we prove that in each of these lattices the partially ordered subset of all those elements which are finitely generated as normal subgroups is a lattice in which infima and suprema of subsets of cardinality $\leq\aleph$, always exist; two infinite distributive identities are also shown to hold.
		In the classes of infinite symmetric groups, their normal subgroups, and their factor groups, we determine those groups which are equivalent in the sense that they may not be distinguished by the solvability of a system of finitely many equations in variables and parameters.
	www.informatik.uni-leipzig.de /~droste/droabal.html (3181 words)

	ATLAS: Linear group L2(16)
		The outer automorphism of order 2 of L2(16) may be achieved by applying this program to the standard generators.
		The outer automorphism of order 4 of L2(16) may be achieved by applying this program to the standard generators.
		(Modulo inner automorphisms, this is the Frobenius automorphism *2.)
	web.mat.bham.ac.uk /atlas/v2.0/lin/L216 (818 words)

	New Results in Combinatorial Designs, Part 1 (Site not responding. Last check: 2007-10-07)
		The distribution of the designs by the orders of their full automorphism groups is as follows: (411,39), (668,78), (312,156), (72,312).
		The distribution of these designs by the orders of their full automorphism groups is as follows: (1635,57), (192,114), (21,171), (39,342), (9,1026).
		There exists a symmetric design with parameters (144,66,30) whose full automorphism group is isomorphic to Aut M_12, the extension of the sporadic simple Mathieu group on 12 letters by an automorphism of order two.
	www.emba.uvm.edu /~dinitz/newresults.part1.html (4096 words)

	Miscellaneous Classification Results for 2-Designs
		The designs, resolvable designs, and their resolutions with a nontrivial automorphism group are available through the links on the table entries.
		Generator permutations for the automorphism group of the design are listed in cycle notation.
		The generator sets given for the automorphism groups are in general not minimal.
	www.tcs.hut.fi /~pkaski/misc-2des.html (317 words)

	Attributes
		for an order o this is an extension of the form O := o[zeta_l] for a prime power l = p^n.
		: the automorphism as an automorphism of O. Note that in general this is not an o-automorphism, as o itself may contain roots of unity.
		Let o be the base ring of the abelian extension A. Then O will be the maximal order of the cyclotomic extension o[zeta_l] as an extensions of Z. The algorithm for the computation of defining extensions will firstly compute a generator a in O such that O(a^(1/l)) equals A(zeta_l).
	wwwmaths.anu.edu.au /research.groups/aat/htmlhelp/text762.htm (1483 words)

	Research (Site not responding. Last check: 2007-10-07)
		We show that a free finitely generated Moufang loop of exponent 3 is finite and that the orders of the finite 2-generated Bol loops of exponent 2 are not bounded.
		The group theoretical problem of the existence of system of representatives T of the subgroup H of G such that T consists of conjugacy classes of involutions leads to the theory Bol loops of exponent 2.
		Due to S. Doro, a group with a splitting automorphism of order $3$ can lead to a group with triality.
	www.math.u-szeged.hu /~nagyg/REGI/Maths (1175 words)

	Kapralov S
		Topalova S., On the order of the automorphism group of 2-(40,10,3) designs, Mathematics and Education in Mathematics (1996), 161-166.
		Topalova S., Hadamard 2-(43,21,10) designs with automorphisms of order 21 and binary self-dual codes of lengths 86 and 88, Proceedings of the Second International Workshop on Optimal Codes and Related Topics, Sozopol, Bulgaria (1998), 193-197.
		Topalova S., Hadamard matrices of order 44 with automorphisms of order 7, Proceedings of the Seventh International Workshop on Algebraic and Combinatorial Coding Theory, Bansko, Bulgaria} (2000), 305-310.
	www.moi.math.bas.bg /~svetlana/publicat.htm (533 words)

	Stefka Bouyuklieva's Home Page
		Self-dual codes with an automorphism of order 7, Mathematics and Education in Mathematics, pp.282-287 (in Bulgarian), 1995.
		On the automorphism group of the extremal singly-even self-dual codes of length 48, Mathematics and Education in Mathematics, pp.
		Binary self-dual codes with an automorphism of order 2, Mathematics and Education in Mathematics, pp.122-127, 1998.
	www.moi.math.bas.bg /~stefka (800 words)

	Table of contents for Library of Congress control number 97028654 (Site not responding. Last check: 2007-10-07)
		Almost regular automorphism of order p: almost nilpotency of p-bounded class 9.
		Almost regular automorphism of order p^n: almost solubility of p^n-bounded derived length 13.
		Automorphism of order p with p^m fixed points: almost nilpotency of m-bounded class Bibliography Index.
	www.loc.gov /catdir/toc/cam027/97028654.html (111 words)

	Isomorphism testing and Standard Presentations
		While it is difficult to state very firm guidelines for the performance of the algorithm, our experience suggests that the difficulty of deciding isomorphism between p-groups is governed by their Frattini rank and is most practical for p-groups of rank at most 5.
		The order of a group is not a useful guide to the difficulty of the computation.
		The function returns the automorphism group of G as a group of type GrpAuto.
	www.math.lsu.edu /magma/text336.htm (682 words)

	Publications by Dietrich Kuske 1999 (Site not responding. Last check: 2007-10-07)
		This paper deals with the automorphism group of the partial order of finite traces.
		Restricting to finite dependence alphabets, the automorphism groups are profinite and possess only finitely many simple composition factors.
		Finally, we show that the partial order associated with the Rado graph as dependence alphabet is not homogeneous thereby answering an open question from [BCS93].
	www.informatik.uni-leipzig.de /~kuske/pub1999.html (248 words)

	Hyperovals: Section 4
		The hyperovals in the Desarguesian planes of orders 2, 4 and 8 are all hyperconics
		Also in 1978, Korchmáros [Ko78] independently gave a constructive proof of this result and showed that the Lunelli-Sce hyperoval is the unique irregular hyperoval admitting a transitive automorphism group (and that the only hyperconics admitting such a group are those of orders 2 and 4).
		In 1991, O'Keefe and Penttila [OKPe92] by means of a detailed investigation of the divisibility properties of the orders of automorphism groups of hypothetical hyperovals in this plane, discovered a new hyperoval.
	www-math.cudenver.edu /~wcherowi/research/hyperoval/hysect4.htm (906 words)

	[No title]
		In order to apply the theorem we are generally faced* * to two main difficulties, namely, to show that the nerve of L(S,f)(X) is homotopy equivalen* *t to X and to show that F(S,f)(X) is a saturated fusion system.
		The outer automorphism group of X(G(m, r, n)) is isomorphic to A(m, r, n)\Zxp* A(m, 1, n) except in the cases (m, r, n) 2 {(2, 1, 2), (4, 2, 2), (3, 3, 3), (2, 2, 4)} [5 *2, x6] [48, 7.14].
		Assume that ff represents an element of finite order r in Out(X), with r prim* *e to p, and X is a connected p-compact group.
	hopf.math.purdue.edu /Broto-Moller/Chev.txt (15825 words)

	Generating p-groups
		The input to the algorithm is a p-group G. The output is a sequence of p-group generation processes: each process provides access to a power-conjugate presentation for a descendant which satisfies chosen parameters and a description of the automorphism group of the descendant.
		Each process provides access to a pcp for a descendant of G and to a sequence of generators for the automorphism group of this descendant.
		If the supplied description of the automorphism group is a PAG-generating sequence which ascends the automorphism group via a polycyclic series with prime factors, set PAGSequence true.
	www.math.uiuc.edu /Software/magma/text233.html (1457 words)

	Diagonally Cyclic Latin Squares (Site not responding. Last check: 2007-10-07)
		A latin square of order $n$ possessing a cyclic automorphism of order $n$ is said to be diagonally cyclic because its entries occur in cyclic order down each broken diagonal.
		An explicit construction is given for a latin square of any odd order.
		The square is conjectured to be $N_\infty$ and this has been confirmed up to order 10000 by computer.
	cs.anu.edu.au /~Ian.Wanless/abstracts/diagcyc.html (242 words)

	Michael Muskulus' Homepage (Site not responding. Last check: 2007-10-07)
		Are the automorphism groups of the full 2-shift and the full 3-shift isomorphic as groups?
		Or consider the earlier question of F. Rhodes: Is every automorphism of the full shift a composition of powers of the shift map and involutions (i.e., automorphism with order two)?
		Prove that: The optimal upper bound for the largest minimal period of a periodic point of a sup-nonexpansive map f: D->D where D is an arbitrary subset of R^n is 2^n.
	www.math.leidenuniv.nl /~muskulus/questions.html (233 words)

	index3 (Site not responding. Last check: 2007-10-07)
		The Automorphism groups of the integer octonions are studied.
		(7) of order 1344 is the nonsplit extension of the elementary
		(7) order 1344 which is the split extension of the elementary Abelian group of order 8 by
	www.gantep.edu.tr /~koc/index3.htm (149 words)

	Order Functions (Site not responding. Last check: 2007-10-07)
		Unless the order is already known, each of the functions in this family will create a faithful permutation representation of the group of automorphisms in order to compute the order.
		The order of the group of automorphisms A, returned as an integer.
		The order of the outer automorphism group associated with the group of automorphisms A. Example
	wwwmaths.anu.edu.au /research.groups/aat/htmlhelp/text366.htm (122 words)

	AMCA: Binary self-dual codes having an automorphism of order $p^2$ by Nikiolay Yankov (Site not responding. Last check: 2007-10-07)
		Using this method, we classify the self-dual [50, 25, 10] and [52, 26, 10] codes with an automorphism of order 25.
		We use the same method to construct the self-dual [36, 18, 8] codes with an automorphism of order 9.
		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/l/m/77.htm (157 words)

	GAP Manual: 84 Coxeter cosets
		A Coxeter coset is the coset WF_0 in the group of automorphisms of V, generated by W and F_0.
		The nontrivial cases to consider are (the order of F is written as left exponent to the type): ^2A_n, ^2D_n, ^3D_4, ^2E_6 and ^2I_2(2k+1).
		The corresponding eigenvalues, sorted in order of increasing degrees of the f_i are called the factors of F_0 acting on V.
	www.mcs.kent.edu /system/documentation/gap/CHAP084.htm (4610 words)

	GAP Manual: 86 Coxeter cosets (Site not responding. Last check: 2007-10-07)
		The nontrivial cases to consider are (the order of F is written as left exponent to the type):
		is an automorphism of finite order of V.
		to determine the order of elements in the coset).
	www.institut.math.jussieu.fr /~jmichel/htm/CHAP086.htm (2690 words)

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