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Hamiltonian cycles in (2,3,c)-circulant digraphs [article]

Dave Witte Morris, Joy Morris, Kerri Webb
<span title="2006-09-30">2006</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Let D be the circulant digraph with n vertices and connection set 2,3,c. (Assume D is loopless and has outdegree 3.) Work of S.C.Locke and D.Witte implies that if n is a multiple of 6, c is either (n/2) + 2 or (n/2) + 3, and c is even, then D does not have a hamiltonian cycle. For all other cases, we construct a hamiltonian cycle in D.
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/math/0610010v1">arXiv:math/0610010v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/z62hfyf7wzcqja56xx6kirjyxu">fatcat:z62hfyf7wzcqja56xx6kirjyxu</a> </span>
<a target="_blank" rel="noopener" href="https://archive.org/download/arxiv-math0610010/math0610010.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> File Archive [PDF] </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/math/0610010v1" title="arxiv.org access"> <button class="ui compact blue labeled icon button serp-button"> <i class="file alternate outline icon"></i> arxiv.org </button> </a>

What is a superrigid subgroup? [article]

Dave Witte Morris
<span title="2007-12-14">2007</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
This is an expository paper. It is well known that a linear transformation can be defined to have any desired action on a basis. From this fact, one can show that every group homomorphism from Z^k to R^d extends to a homomorphism from R^k to R^d, and we will see other examples of discrete subgroups H of connected groups G, such that the homomorphisms defined on H can ("almost") be extended to homomorphisms defined on all of G. This is related to a very classical topic in geometry, the study of linkages.
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/0712.2299v1">arXiv:0712.2299v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/zanjpudujfcm5pgdpp2pb4bswa">fatcat:zanjpudujfcm5pgdpp2pb4bswa</a> </span>
<a target="_blank" rel="noopener" href="https://archive.org/download/arxiv-0712.2299/0712.2299.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> File Archive [PDF] </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/0712.2299v1" title="arxiv.org access"> <button class="ui compact blue labeled icon button serp-button"> <i class="file alternate outline icon"></i> arxiv.org </button> </a>

Isomorphisms of Cayley graphs on nilpotent groups [article]

Dave Witte Morris, Joy Morris, Gabriel Verret
<span title="2016-03-11">2016</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Let S be a finite generating set of a torsion-free, nilpotent group G. We show that every automorphism of the Cayley graph Cay(G;S) is affine. (That is, every automorphism of the graph is obtained by composing a group automorphism with multiplication by an element of the group.) More generally, we show that if Cay(G;S) and Cay(G';S') are connected Cayley graphs of finite valency on two nilpotent groups G and G', then every isomorphism from Cay(G;S) to Cay(G';S') factors through to a
more &raquo; ... affine map from G/N to G'/N', where N and N' are the torsion subgroups of G and G', respectively. For the special case where the groups are abelian, these results were previously proved by A.A.Ryabchenko and C.Loeh, respectively.
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1603.01883v2">arXiv:1603.01883v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/adqzspvxjfeobcmp3i3hs2mi44">fatcat:adqzspvxjfeobcmp3i3hs2mi44</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20200902050213/https://arxiv.org/pdf/1603.01883v2.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/84/db/84dba8a14f1b6b21d4673e8fd9fbebc755d18fc8.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1603.01883v2" title="arxiv.org access"> <button class="ui compact blue labeled icon button serp-button"> <i class="file alternate outline icon"></i> arxiv.org </button> </a>

Cayley graphs of order 16p are hamiltonian [article]

Stephen J. Curran, Dave Witte Morris, Joy Morris
<span title="2011-04-04">2011</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Suppose G is a finite group, such that |G| = 16p, where p is prime. We show that if S is any generating set of G, then there is a hamiltonian cycle in the corresponding Cayley graph Cay(G;S).
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1104.0081v2">arXiv:1104.0081v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/f5j2pdbgvbcpllig55xtneibde">fatcat:f5j2pdbgvbcpllig55xtneibde</a> </span>
<a target="_blank" rel="noopener" href="https://archive.org/download/arxiv-1104.0081/1104.0081.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> File Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/b8/ea/b8eacfe7fb769ca1e4d9a1f1443db63da7c4b219.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1104.0081v2" title="arxiv.org access"> <button class="ui compact blue labeled icon button serp-button"> <i class="file alternate outline icon"></i> arxiv.org </button> </a>

Hamiltonian cycles in (2,3,c)-circulant digraphs

Dave Witte Morris, Joy Morris, Kerri Webb
<span title="">2009</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/civgv5utqzhu7aj6voo6vc5vx4" style="color: black;">Discrete Mathematics</a> </i> &nbsp;
Let D be the circulant digraph with n vertices and connection set {2, 3, c}. (Assume D is loopless and has outdegree 3.) Work of S. C. Locke and D. Witte implies that if n is a multiple of 6, c ∈ {(n/2) + 2, (n/2) + 3}, and c is even, then D does not have a hamiltonian cycle. For all other cases, we construct a hamiltonian cycle in D.
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.disc.2009.01.001">doi:10.1016/j.disc.2009.01.001</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/mipjzyf4qzfylgbxx7brh4i4oy">fatcat:mipjzyf4qzfylgbxx7brh4i4oy</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20190318073404/https://core.ac.uk/download/pdf/82366245.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/65/7c/657c8d6bb7a6950c395337e8d68c6563858144c5.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.disc.2009.01.001"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> elsevier.com </button> </a>

On colour-preserving automorphisms of Cayley graphs [article]

Ademir Hujdurović, Klavdija Kutnar, Dave Witte Morris, Joy Morris
<span title="2015-03-26">2015</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Morris, J. Morris  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1411.6732v2">arXiv:1411.6732v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/5fk4xdykkremjio2nc4lfqiymm">fatcat:5fk4xdykkremjio2nc4lfqiymm</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20200908130238/https://arxiv.org/pdf/1411.6732v2.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/97/b6/97b643ea96366a637610a1388bf66e91e2b48960.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1411.6732v2" title="arxiv.org access"> <button class="ui compact blue labeled icon button serp-button"> <i class="file alternate outline icon"></i> arxiv.org </button> </a>

Cayley graphs of order 16p are hamiltonian

Stephen J. Curran, Dave Witte Morris, Joy Morris
<span title="2012-03-21">2012</span> <i title="University of Primorska Press"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/gvgtrk4iyvatbibvfwr3uifrji" style="color: black;">Ars Mathematica Contemporanea</a> </i> &nbsp;
Suppose G is a finite group, such that |G| = 16p, where p is prime. We show that if S is any generating set of G, then there is a hamiltonian cycle in the corresponding Cayley graph Cay(G; S).
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.26493/1855-3974.207.8e0">doi:10.26493/1855-3974.207.8e0</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/oa6q3eolhrhydbdutgww3o2yda">fatcat:oa6q3eolhrhydbdutgww3o2yda</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170705054646/http://amc-journal.eu/index.php/amc/article/download/207/162/" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/fe/c0/fec01355077de26c613d507843234ff172084ca0.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.26493/1855-3974.207.8e0"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="unlock alternate icon" style="background-color: #fb971f;"></i> Publisher / doi.org </button> </a>

Horospherical limit points of S-arithmetic groups [article]

Dave Witte Morris, Kevin Wortman
<span title="2013-09-27">2013</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Suppose Gamma is an S-arithmetic subgroup of a connected, semisimple algebraic group G over a global field Q (of any characteristic). It is well known that Gamma acts by isometries on a certain CAT(0) metric space X_S that is a Cartesian product of Euclidean buildings and Riemannian symmetric spaces. For a point p on the visual boundary of X_S, we show there exists a horoball based at p that is disjoint from some Gamma-orbit in X_S if and only if p lies on the boundary of a certain type of flat
more &raquo; ... in X_S that we call "Q-good." This generalizes a theorem of G.Avramidi and D.W.Morris that characterizes the horospherical limit points for the action of an arithmetic group on its associated symmetric space.
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1309.7113v1">arXiv:1309.7113v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/udterowcanahbcmoe3nmah3jqa">fatcat:udterowcanahbcmoe3nmah3jqa</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20200925033240/https://arxiv.org/pdf/1309.7113v1.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/81/e5/81e5e207a3aae45fd8050da0c340b5639ca5714c.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1309.7113v1" title="arxiv.org access"> <button class="ui compact blue labeled icon button serp-button"> <i class="file alternate outline icon"></i> arxiv.org </button> </a>

Dani's Work on Dynamical Systems on Homogeneous Spaces [article]

Dave Witte Morris
<span title="2013-07-25">2013</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
In honor of S.G.Dani's 65th birthday, we describe some of his many contributions to the theory and applications of dynamical systems on homogeneous spaces, with emphasis on unipotent flows.
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1307.6893v1">arXiv:1307.6893v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/bxlzx6wsg5dxxhqjo6tdafhola">fatcat:bxlzx6wsg5dxxhqjo6tdafhola</a> </span>
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On colour-preserving automorphisms of Cayley graphs

Ademir Hujdurović, Klavdija Kutnar, Dave Witte Morris, Joy Morris
<span title="2015-10-20">2015</span> <i title="University of Primorska Press"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/gvgtrk4iyvatbibvfwr3uifrji" style="color: black;">Ars Mathematica Contemporanea</a> </i> &nbsp;
Morris, J. Morris  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.26493/1855-3974.771.9b3">doi:10.26493/1855-3974.771.9b3</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/wbhecoq4tffnpb5sbpbyvlkvba">fatcat:wbhecoq4tffnpb5sbpbyvlkvba</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20190224003738/http://pdfs.semanticscholar.org/4ff4/fedea58ab036fd29cb4aca015aebdfab6aaf.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/4f/f4/4ff4fedea58ab036fd29cb4aca015aebdfab6aaf.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.26493/1855-3974.771.9b3"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="unlock alternate icon" style="background-color: #fb971f;"></i> Publisher / doi.org </button> </a>

IVOA Recommendation: VOSpace specification Version 1.15 [article]

Matthew Graham, Dave Morris, Guy Rixon
<span title="2011-10-03">2011</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
VOSpace is the IVOA interface to distributed storage. This version extends the existing VOSpace 1.0 (SOAP-based) specification to support containers, links between individual VOSpace instances, third party APIs, and a find mechanism. Note, however, that VOSpace-1.0 compatible clients will not work with this new version of the interface.
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1110.0508v1">arXiv:1110.0508v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/gox5ub4ecbbj3o54mgh432zvbm">fatcat:gox5ub4ecbbj3o54mgh432zvbm</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20200824110554/https://arxiv.org/ftp/arxiv/papers/1110/1110.0508.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/b0/fe/b0fee691a8281074c7778bde66d39de19db09d71.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1110.0508v1" title="arxiv.org access"> <button class="ui compact blue labeled icon button serp-button"> <i class="file alternate outline icon"></i> arxiv.org </button> </a>

Hamiltonian paths in m x n projective checkerboards [article]

Dallan McCarthy, Dave Witte Morris
<span title="2016-07-14">2016</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
For any two squares A and B of an m x n checkerboard, we determine whether it is possible to move a checker through a route that starts at A, ends at B, and visits each square of the board exactly once. Each step of the route moves to an adjacent square, either to the east or to the north, and may step off the edge of the board in a manner corresponding to the usual construction of a projective plane by applying a twist when gluing opposite sides of a rectangle. This generalizes work of M.H.Forbush et al. for the special case where m = n.
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1607.04001v1">arXiv:1607.04001v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/j4zvndt3bjhvnfbhc5u36a2hle">fatcat:j4zvndt3bjhvnfbhc5u36a2hle</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20200831182525/https://arxiv.org/pdf/1607.04001v1.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/f0/80/f080583cd340dd7532bdd2308e6246cc14fc8a23.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1607.04001v1" title="arxiv.org access"> <button class="ui compact blue labeled icon button serp-button"> <i class="file alternate outline icon"></i> arxiv.org </button> </a>

Infinitely many nonsolvable groups whose Cayley graphs are hamiltonian [article]

Dave Witte Morris
<span title="2015-07-17">2015</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Appendix: Notes to aid the referee Infinitely many nonsolvable groups whose Cayley graphs are hamiltonian by Dave Witte Morris A.1. Since gcd(|A 5 |, p) = 1, we have g ∈ g for every g ∈ A 5 × Z p .  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1507.04973v1">arXiv:1507.04973v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/chu7idrz55fnjdq7kofane7ixu">fatcat:chu7idrz55fnjdq7kofane7ixu</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20200826041218/https://arxiv.org/pdf/1507.04973v1.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/46/8f/468fe13cc3390a531ce9d61b97028182af25b0fe.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1507.04973v1" title="arxiv.org access"> <button class="ui compact blue labeled icon button serp-button"> <i class="file alternate outline icon"></i> arxiv.org </button> </a>

On Cayley digraphs that do not have hamiltonian paths [article]

Dave Witte Morris
<span title="2013-06-23">2013</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Morris)  ...  The following known result handles the case where G is nilpotent: Theorem 4.5 (Morris [6]). Assume G is nilpotent, and S generates G.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1306.5443v1">arXiv:1306.5443v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/7xau6xkgq5enbdxgea6feybefy">fatcat:7xau6xkgq5enbdxgea6feybefy</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20191018060249/https://arxiv.org/pdf/1306.5443v1.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/2c/3e/2c3ef7f8671bd7aa3f3cea03d0f451ad6047d8e5.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1306.5443v1" title="arxiv.org access"> <button class="ui compact blue labeled icon button serp-button"> <i class="file alternate outline icon"></i> arxiv.org </button> </a>

On hamiltonian cycles in Cayley graphs of order pqrs [article]

Dave Witte Morris
<span title="2021-07-30">2021</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Theorem 2.2 ([1, p. 257], Durnberger [2] , Morris [8] ). Let A be a generating set of a nontrivial finite group G, and assume that |G| is odd.  ... 
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