<html><head><title>The GAP 4 Reference Manual - Index M</title></head> <body text="#000000" bgcolor="#ffffff"> <h1>The <font face="Gill Sans,Helvetica,Arial">GAP</font> 4 Reference Manual - Index M</h1> <p> <a href="theindex.htm">_</A> <a href="indxA.htm">A</A> <a href="indxB.htm">B</A> <a href="indxC.htm">C</A> <a href="indxD.htm">D</A> <a href="indxE.htm">E</A> <a href="indxF.htm">F</A> <a href="indxG.htm">G</A> <a href="indxH.htm">H</A> <a href="indxI.htm">I</A> <a href="indxJ.htm">J</A> <a href="indxK.htm">K</A> <a href="indxL.htm">L</A> <a href="indxM.htm">M</A> <a href="indxN.htm">N</A> <a href="indxO.htm">O</A> <a href="indxP.htm">P</A> <a href="indxQ.htm">Q</A> <a href="indxR.htm">R</A> <a href="indxS.htm">S</A> <a href="indxT.htm">T</A> <a href="indxU.htm">U</A> <a href="indxV.htm">V</A> <a href="indxW.htm">W</A> <a href="indxX.htm">X</A> <a href="indxY.htm">Y</A> <a href="indxZ.htm">Z</A> <dt>m_N <a href="CHAP018.htm#I30">18.4</a> <dt>Macintosh <a href="CHAP073.htm#I8">73.14</a> <a href="CHAP073.htm#I11">73.15</a> <dt>MacOS <a href="CHAP073.htm#I9">73.15</a> <dt>Magma <a href="CHAP033.htm#SSEC002.1">33.2.1</a> <dt>Magma Categories <a href="CHAP033.htm#SECT001">33.1</a> <dt>Magma Generation <a href="CHAP033.htm#SECT002">33.2</a> <dt>Magma Homomorphisms <a href="CHAP031.htm#SECT007">31.7</a> <dt>Magma Rings <a href="CHAP063.htm">63.0</a> <dt>Magma Rings modulo Relations <a href="CHAP063.htm#SECT004">63.4</a> <dt>Magma Rings modulo the Span of a Zero Element <a href="CHAP063.htm#SECT005">63.5</a> <dt>MagmaByGenerators <a href="CHAP033.htm#SSEC002.4">33.2.4</a> <dt>MagmaByMultiplicationTable <a href="CHAP033.htm#SSEC003.1">33.3.1</a> <dt>MagmaElement <a href="CHAP033.htm#SSEC003.4">33.3.4</a> <dt>MagmaHomomorphismByFunctionNC <a href="CHAP031.htm#SSEC007.2">31.7.2</a> <dt>MagmaRingModuloSpanOfZero <a href="CHAP063.htm#SSEC005.3">63.5.3</a> <dt>Magmas <a href="CHAP033.htm">33.0</a> <dt>Magmas Defined by Multiplication Tables <a href="CHAP033.htm#SECT003">33.3</a> <dt>MagmaWithInverses <a href="CHAP033.htm#SSEC002.3">33.2.3</a> <dt>MagmaWithInversesByGenerators <a href="CHAP033.htm#SSEC002.6">33.2.6</a> <dt>MagmaWithInversesByMultiplicationTable <a href="CHAP033.htm#SSEC003.3">33.3.3</a> <dt>MagmaWithOne <a href="CHAP033.htm#SSEC002.2">33.2.2</a> <dt>MagmaWithOneByGenerators <a href="CHAP033.htm#SSEC002.5">33.2.5</a> <dt>MagmaWithOneByMultiplicationTable <a href="CHAP033.htm#SSEC003.2">33.3.2</a> <dt>Main Loop <a href="CHAP006.htm#SECT001">6.1</a> <dt>Main Loop and Break Loop <a href="CHAP006.htm">6.0</a> <dt>MakeConfluent <a href="CHAP036.htm#SSEC001.11">36.1.11</a> <dt>MakeImmutable <a href="CHAP012.htm#SSEC006.4">12.6.4</a> <dt>MakeReadOnlyGlobal <a href="CHAP004.htm#SSEC009.2">4.9.2</a> <dt>MakeReadWriteGlobal <a href="CHAP004.htm#SSEC009.3">4.9.3</a> <dt>Making transformation semigroups <a href="CHAP049.htm#SECT001">49.1</a> <dt>Manual Conventions <a href="CHAP001.htm#SECT001">1.1</a> <dt>map, parametrized <a href="CHAP071.htm#I6">71.3</a> <dt>MappedWord <a href="CHAP034.htm#SSEC003.1">34.3.1</a> <dt>MappingByFunction <a href="CHAP031.htm#SSEC001.2">31.1.2</a> <dt>MappingPermListList <a href="CHAP040.htm#SSEC004.3">40.4.3</a> <dt>Mappings <a href="CHAP031.htm">31.0</a> <dt>Mappings that Respect Addition <a href="CHAP031.htm#SECT009">31.9</a> <dt>Mappings that Respect Multiplication <a href="CHAP031.htm#SECT008">31.8</a> <dt>Mappings which are Compatible with Algebraic Structures <a href="CHAP031.htm#SECT006">31.6</a> <dt>maps <a href="CHAP071.htm#I0">71.0</a> <dt>Maps Concerning Character Tables <a href="CHAP071.htm">71.0</a> <dt>MarksTom <a href="CHAP068.htm#SSEC007.1">68.7.1</a> <dt>MatAlgebra <a href="CHAP060.htm#SSEC004.4">60.4.4</a> <dt>MatClassMultCoeffsCharTable <a href="CHAP069.htm#SSEC010.8">69.10.8</a> <dt>MathieuGroup <a href="CHAP048.htm#SSEC001.9">48.1.9</a> <dt>MatLieAlgebra <a href="CHAP061.htm#SSEC002.4">61.2.4</a> <dt>Matrices <a href="CHAP024.htm">24.0</a> <dt>Matrices as Basis of a Row Space <a href="CHAP024.htm#SECT010">24.10</a> <dt>Matrices as Linear Mappings <a href="CHAP024.htm#SECT012">24.12</a> <dt>Matrices over Finite Fields <a href="CHAP024.htm#SECT013">24.13</a> <dt>Matrices Representing Linear Equations and the Gaussian Algorithm <a href="CHAP024.htm#SECT006">24.6</a> <dt>matrices, commutator <a href="CHAP024.htm#SSEC002.11">24.2.11</a> <dt>matrix automorphisms <a href="CHAP071.htm#I2">71.1</a> <dt>Matrix Constructions <a href="CHAP024.htm#SECT004">24.4</a> <dt>Matrix Groups <a href="CHAP042.htm">42.0</a> <dt>Matrix Groups in Characteristic 0 <a href="CHAP042.htm#SECT005">42.5</a> <dt>matrix spaces <a href="CHAP059.htm#I1">59.8</a> <dt>MatrixAlgebra <a href="CHAP060.htm#SSEC004.4">60.4.4</a> <dt>MatrixAutomorphisms <a href="CHAP069.htm#SSEC020.1">69.20.1</a> <dt>MatrixByBlockMatrix <a href="CHAP024.htm#SSEC015.3">24.15.3</a> <dt>MatrixLieAlgebra <a href="CHAP061.htm#SSEC002.4">61.2.4</a> <dt>MatrixOfAction <a href="CHAP060.htm#SSEC010.15">60.10.15</a> <dt>MatScalarProducts <a href="CHAP070.htm#SSEC008.6">70.8.6</a> <dt>MatTom <a href="CHAP068.htm#SSEC007.10">68.7.10</a> <dt>MaximalAbelianQuotient <a href="CHAP037.htm#SSEC018.4">37.18.4</a> <dt>MaximalBlocks <a href="CHAP039.htm#SSEC010.2">39.10.2</a> <dt>MaximalNormalSubgroups <a href="CHAP037.htm#SSEC019.8">37.19.8</a> <dt>MaximalSubgroupClassReps <a href="CHAP037.htm#SSEC019.5">37.19.5</a> <dt>MaximalSubgroups <a href="CHAP037.htm#SSEC019.6">37.19.6</a> <dt>MaximalSubgroups, for groups with pcgs <a href="CHAP043.htm#I8">43.16</a> <dt>MaximalSubgroupsLattice <a href="CHAP037.htm#SSEC020.3">37.20.3</a> <dt>MaximalSubgroupsTom <a href="CHAP068.htm#SSEC009.12">68.9.12</a> <dt>Maximum <a href="CHAP021.htm#SSEC020.12">21.20.12</a> <dt>MaximumList <a href="CHAP021.htm#SSEC020.14">21.20.14</a> <dt>MeatAxe Modules <a href="CHAP067.htm#SECT001">67.1</a> <dt>MeetEquivalenceRelations <a href="CHAP032.htm#SSEC006.3">32.6.3</a> <dt>MeetMaps <a href="CHAP071.htm#SSEC003.8">71.3.8</a> <dt>Membership Test for Collections <a href="CHAP028.htm#SECT005">28.5</a> <dt>Membership Test for Lists <a href="CHAP021.htm#SECT008">21.8</a> <dt>MemoryUsage <a href="CHAP012.htm#SSEC008.4">12.8.4</a> <dt>Minimal Nonmonomial Groups <a href="CHAP072.htm#SECT004">72.4</a> <dt>Minimal Polynomials <a href="CHAP064.htm#SECT008">64.8</a> <dt>MinimalElementCosetStabChain <a href="CHAP041.htm#SSEC009.12">41.9.12</a> <dt>MinimalGeneratingSet <a href="CHAP037.htm#SSEC022.3">37.22.3</a> <dt>MinimalGeneratingSet, for groups with pcgs <a href="CHAP043.htm#I10">43.16</a> <dt>MinimalNonmonomialGroup <a href="CHAP072.htm#SSEC004.2">72.4.2</a> <dt>MinimalNormalSubgroups <a href="CHAP037.htm#SSEC019.9">37.19.9</a> <dt>MinimalPolynomial <a href="CHAP064.htm#SSEC008.1">64.8.1</a> <dt>MinimalPolynomial, over a field <a href="CHAP056.htm#SSEC003.2">56.3.2</a> <dt>MinimalPolynomial, over a ring <a href="CHAP064.htm#I0">64.8</a> <dt>MinimalStabChain <a href="CHAP041.htm#SSEC007.5">41.7.5</a> <dt>MinimalSupergroupsLattice <a href="CHAP037.htm#SSEC020.4">37.20.4</a> <dt>MinimalSupergroupsTom <a href="CHAP068.htm#SSEC009.13">68.9.13</a> <dt>MinimizedBombieriNorm <a href="CHAP064.htm#SSEC011.6">64.11.6</a> <dt>Minimum <a href="CHAP021.htm#SSEC020.13">21.20.13</a> <dt>MinimumList <a href="CHAP021.htm#SSEC020.14">21.20.14</a> <dt>MinusCharacter <a href="CHAP071.htm#SSEC004.5">71.4.5</a> <dt>Miscellaneous <a href="CHAP015.htm#SECT006">15.6</a> <dt>Miscellaneous Name Changes or Removed Names <a href="CHAP075.htm#SECT004">75.4</a> <dt>mod <a href="CHAP004.htm#SSEC012.1">4.12.1</a> <dt>mod, arithmetic operators <a href="CHAP004.htm#I20">4.12</a> <dt>mod, for character tables <a href="CHAP069.htm#I9">69.7</a> <dt>mod, Integers <a href="CHAP014.htm#I11">14.4</a> <dt>mod, Laurent polynomials <a href="CHAP064.htm#SSEC002.2">64.2.2</a> <dt>mod, lists <a href="CHAP021.htm#I13">21.14</a> <dt>mod, rationals <a href="CHAP004.htm#I23">4.12</a> <dt>mod, residue class rings <a href="CHAP014.htm#I10">14.4</a> <dt>modular inverse <a href="CHAP004.htm#I25">4.12</a> <dt>modular remainder <a href="CHAP004.htm#I24">4.12</a> <dt>modular roots <a href="CHAP015.htm#I17">15.3</a> <dt>Module Constructions <a href="CHAP067.htm#SECT002">67.2</a> <dt>Module Homomorphisms <a href="CHAP067.htm#SECT008">67.8</a> <dt>ModuleByRestriction <a href="CHAP060.htm#SSEC010.21">60.10.21</a> <dt>ModuleOfExtension <a href="CHAP044.htm#SSEC008.7">44.8.7</a> <dt>Modules <a href="CHAP055.htm">55.0</a> <dt>Modules over Lie Algebras and Their Cohomology <a href="CHAP061.htm#SECT012">61.12</a> <dt>Modules over Semisimple Lie Algebras <a href="CHAP061.htm#SECT013">61.13</a> <dt>modulo <a href="CHAP004.htm#I21">4.12</a> <dt>modulo, arithmetic operators <a href="CHAP004.htm#I22">4.12</a> <dt>modulo, for pcgs <a href="CHAP043.htm#SSEC009.5">43.9.5</a> <dt>modulo, residue class rings <a href="CHAP014.htm#SSEC004.1">14.4.1</a> <dt>ModuloPcgs <a href="CHAP043.htm#SSEC009.1">43.9.1</a> <dt>MoebiusMu <a href="CHAP015.htm#SSEC004.3">15.4.3</a> <dt>MoebiusTom <a href="CHAP068.htm#SSEC007.11">68.7.11</a> <dt>Molien Series <a href="CHAP070.htm#SECT012">70.12</a> <dt>MolienSeries <a href="CHAP070.htm#SSEC012.1">70.12.1</a> <dt>MolienSeriesInfo <a href="CHAP070.htm#SSEC012.2">70.12.2</a> <dt>MolienSeriesWithGivenDenominator <a href="CHAP070.htm#SSEC012.4">70.12.4</a> <dt>Monoid <a href="CHAP050.htm#">50.0</a> <a href="CHAP050.htm#">50.0</a> <a href="CHAP050.htm#">50.0</a> <dt>MonoidByGenerators <a href="CHAP050.htm#">50.0</a> <a href="CHAP050.htm#">50.0</a> <dt>MonoidByMultiplicationTable <a href="CHAP050.htm#">50.0</a> <dt>MonoidOfRewritingSystem <a href="CHAP051.htm#SSEC005.5">51.5.5</a> <dt>Monoids <a href="CHAP050.htm">50.0</a> <dt>Monomial Orderings <a href="CHAP064.htm#SECT016">64.16</a> <dt>MonomialComparisonFunction <a href="CHAP064.htm#SSEC016.5">64.16.5</a> <dt>MonomialExtGrlexLess <a href="CHAP064.htm#SSEC016.14">64.16.14</a> <dt>MonomialExtrepComparisonFun <a href="CHAP064.htm#SSEC016.6">64.16.6</a> <dt>MonomialGrevlexOrdering <a href="CHAP064.htm#SSEC016.9">64.16.9</a> <dt>MonomialGrlexOrdering <a href="CHAP064.htm#SSEC016.8">64.16.8</a> <dt>Monomiality Questions <a href="CHAP072.htm">72.0</a> <dt>MonomialLexOrdering <a href="CHAP064.htm#SSEC016.7">64.16.7</a> <dt>MonomialTotalDegreeLess <a href="CHAP075.htm#I21">75.4</a> <dt>monomorphisms, find all <a href="CHAP038.htm#I7">38.9</a> <dt>MorClassLoop <a href="CHAP038.htm#SSEC009.4">38.9.4</a> <dt>More about Boolean Lists <a href="CHAP022.htm#SECT004">22.4</a> <dt>More About Global Variables <a href="CHAP004.htm#SECT009">4.9</a> <dt>More about Tables of Marks <a href="CHAP068.htm#SECT001">68.1</a> <dt>MostFrequentGeneratorFpGroup <a href="CHAP045.htm#SSEC005.8">45.5.8</a> <dt>Moved Points of Permutations <a href="CHAP040.htm#SECT002">40.2</a> <dt>MovedPoints <a href="CHAP040.htm#SSEC002.3">40.2.3</a> <dt>MTX.BasesCompositionSeries <a href="CHAP067.htm#SSEC006.9">67.6.9</a> <dt>MTX.BasesMaximalSubmodules <a href="CHAP067.htm#SSEC006.5">67.6.5</a> <dt>MTX.BasesMinimalSubmodules <a href="CHAP067.htm#SSEC006.4">67.6.4</a> <dt>MTX.BasesMinimalSupermodules <a href="CHAP067.htm#SSEC006.8">67.6.8</a> <dt>MTX.BasesSubmodules <a href="CHAP067.htm#SSEC006.3">67.6.3</a> <dt>MTX.BasisInOrbit <a href="CHAP067.htm#SSEC009.4">67.9.4</a> <dt>MTX.BasisRadical <a href="CHAP067.htm#SSEC006.6">67.6.6</a> <dt>MTX.BasisSocle <a href="CHAP067.htm#SSEC006.7">67.6.7</a> <dt>MTX.CollectedFactors <a href="CHAP067.htm#SSEC006.11">67.6.11</a> <dt>MTX.CompositionFactors <a href="CHAP067.htm#SSEC006.10">67.6.10</a> <dt>MTX.DegreeSplittingField <a href="CHAP067.htm#SSEC005.3">67.5.3</a> <dt>MTX.Dimension <a href="CHAP067.htm#SSEC004.2">67.4.2</a> <dt>MTX.Distinguish <a href="CHAP067.htm#SSEC008.5">67.8.5</a> <dt>MTX.Field <a href="CHAP067.htm#SSEC004.3">67.4.3</a> <dt>MTX.Generators <a href="CHAP067.htm#SSEC004.1">67.4.1</a> <dt>MTX.Homomorphism <a href="CHAP067.htm#SSEC008.3">67.8.3</a> <dt>MTX.Homomorphisms <a href="CHAP067.htm#SSEC008.4">67.8.4</a> <dt>MTX.InducedAction <a href="CHAP067.htm#SSEC007.5">67.7.5</a> <dt>MTX.InducedActionFactorMatrix <a href="CHAP067.htm#SSEC007.4">67.7.4</a> <dt>MTX.InducedActionFactorModule <a href="CHAP067.htm#SSEC007.3">67.7.3</a> <dt>MTX.InducedActionMatrix <a href="CHAP067.htm#SSEC007.4">67.7.4</a> <dt>MTX.InducedActionMatrixNB <a href="CHAP067.htm#SSEC007.4">67.7.4</a> <dt>MTX.InducedActionSubmodule <a href="CHAP067.htm#SSEC007.2">67.7.2</a> <dt>MTX.InducedActionSubmoduleNB <a href="CHAP067.htm#SSEC007.2">67.7.2</a> <dt>MTX.InvariantBilinearForm <a href="CHAP067.htm#SSEC009.1">67.9.1</a> <dt>MTX.InvariantQuadraticForm <a href="CHAP067.htm#SSEC009.3">67.9.3</a> <dt>MTX.InvariantSesquilinearForm <a href="CHAP067.htm#SSEC009.2">67.9.2</a> <dt>MTX.IsAbsolutelyIrreducible <a href="CHAP067.htm#SSEC005.2">67.5.2</a> <dt>MTX.IsEquivalent <a href="CHAP067.htm#SSEC008.1">67.8.1</a> <dt>MTX.IsIrreducible <a href="CHAP067.htm#SSEC005.1">67.5.1</a> <dt>MTX.Isomorphism <a href="CHAP067.htm#SSEC008.2">67.8.2</a> <dt>MTX.NormedBasisAndBaseChange <a href="CHAP067.htm#SSEC007.1">67.7.1</a> <dt>MTX.OrthogonalSign <a href="CHAP067.htm#SSEC009.5">67.9.5</a> <dt>MTX.ProperSubmoduleBasis <a href="CHAP067.htm#SSEC006.2">67.6.2</a> <dt>MTX.SubGModule <a href="CHAP067.htm#SSEC006.1">67.6.1</a> <dt>MTX.SubmoduleGModule <a href="CHAP067.htm#SSEC006.1">67.6.1</a> <dt>multiplication <a href="CHAP004.htm#SSEC012.1">4.12.1</a> <dt>multiplication, matrices <a href="CHAP024.htm#SSEC002.7">24.2.7</a> <dt>multiplication, matrix and matrix list <a href="CHAP024.htm#SSEC002.13">24.2.13</a> <dt>multiplication, matrix and scalar <a href="CHAP024.htm#SSEC002.4">24.2.4</a> <dt>multiplication, matrix and vector <a href="CHAP024.htm#SSEC002.6">24.2.6</a> <dt>multiplication, operation <a href="CHAP030.htm#SSEC012.1">30.12.1</a> <dt>multiplication, scalar and matrix <a href="CHAP024.htm#SSEC002.4">24.2.4</a> <dt>multiplication, scalar and matrix list <a href="CHAP024.htm#SSEC002.12">24.2.12</a> <dt>multiplication, scalar and vector <a href="CHAP023.htm#SSEC001.4">23.1.4</a> <dt>multiplication, vector and matrix <a href="CHAP024.htm#SSEC002.5">24.2.5</a> <dt>multiplication, vector and matrix list <a href="CHAP024.htm#SSEC002.15">24.2.15</a> <dt>multiplication, vector and scalar <a href="CHAP023.htm#SSEC001.4">23.1.4</a> <dt>multiplication, vectors <a href="CHAP023.htm#SSEC001.5">23.1.5</a> <dt>MultiplicationTable <a href="CHAP033.htm#SSEC003.5">33.3.5</a> <dt>Multiplicative Arithmetic for Lists <a href="CHAP021.htm#SECT014">21.14</a> <dt>Multiplicative Arithmetic Functions <a href="CHAP015.htm#SECT004">15.4</a> <dt>multiplicative order of an integer <a href="CHAP015.htm#I8">15.2</a> <dt>MultiplicativeNeutralElement <a href="CHAP033.htm#SSEC004.10">33.4.10</a> <dt>MultiplicativeZero <a href="CHAP033.htm#SSEC004.11">33.4.11</a> <dt>MultiplicativeZeroOp <a href="CHAP030.htm#SSEC010.4">30.10.4</a> <dt>multiplicity, of constituents of a group character <a href="CHAP070.htm#I25">70.8</a> <dt>Multiplier <a href="CHAP037.htm#I13">37.24</a> <dt>multisets <a href="CHAP021.htm#I19">21.19</a> <dt>Multivariate Polynomials <a href="CHAP064.htm#SECT007">64.7</a> <dt>MultRowVector <a href="CHAP023.htm#SSEC003.3">23.3.3</a> <dt>Murnaghan components <a href="CHAP070.htm#I39">70.11</a> <dt>Mutability and Copyability <a href="CHAP012.htm#SECT006">12.6</a> <dt>Mutability Status and List Arithmetic <a href="CHAP021.htm#SECT015">21.15</a> <dt>Mutable Bases <a href="CHAP059.htm#SECT007">59.7</a> <dt>MutableBasis <a href="CHAP059.htm#SSEC007.2">59.7.2</a> <dt>MutableBasisOfClosureUnderAction <a href="CHAP060.htm#SSEC008.11">60.8.11</a> <dt>MutableBasisOfIdealInNonassociativeAlgebra <a href="CHAP060.htm#SSEC008.13">60.8.13</a> <dt>MutableBasisOfNonassociativeAlgebra <a href="CHAP060.htm#SSEC008.12">60.8.12</a> <dt>MutableCopyMat <a href="CHAP024.htm#SSEC004.13">24.4.13</a> <dt>MutableIdentityMat <a href="CHAP024.htm#SSEC004.11">24.4.11</a> <dt>MutableNullMat <a href="CHAP024.htm#SSEC004.12">24.4.12</a> </dl><p> [<a href="../index.htm">Top</a>] [<a href="chapters.htm">Up</a>]<p> <P> <font face="Gill Sans,Helvetica,Arial">GAP 4 manual<br>December 2008 </font></body></html>
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