Introduction to Groups - The University of Western Australia

Introduction to GroupsAlice NiemeyerThe University ofWestern AustraliaIntroduction to GroupsAlice NiemeyerThe University of Western AustraliaGroupsHexagon SymmetriesPermutationsPermutation GroupsGeneratorsOrbitsCosetsStabilisersOrbit Stabiliser TheoremBase & Stabiliser ChainSeptember 2004

Introduction to GroupsAlice NiemeyerThe University ofWestern AustraliaDefinitionA group (G, ∗) is abelian if the binaryoperation ∗ is commutative.GroupsHexagon SymmetriesPermutationsPermutation GroupsGeneratorsOrbitsCosetsStabilisersOrbit Stabiliser TheoremBase & Stabiliser Chain

ExamplesIntroduction to GroupsAlice NiemeyerThe University ofWestern Australia1. (Z, +), (Q, +), (R, +), (C, +) aregroups.2. (Q\{0}, ·), (R\{0}, ·), (C\{0}, ·) aregroups.3. (R n , +), the set of vectors in R n undervector addition, is a group.GroupsHexagon SymmetriesPermutationsPermutation GroupsGeneratorsOrbitsCosetsStabilisersOrbit Stabiliser TheoremBase & Stabiliser Chain

More ExamplesIntroduction to GroupsAlice NiemeyerThe University ofWestern Australia4. (M n (R), ∗), the set of n × n matriceswith entries in R under matrixmultiplication, is not a group.5. (GL n (R), ∗), the set of invertible n × nmatrices with entries in R under matrixmultiplication, is a group. This group iscalled the general linear group over R.This group is not abelian.GroupsHexagon SymmetriesPermutationsPermutation GroupsGeneratorsOrbitsCosetsStabilisersOrbit Stabiliser TheoremBase & Stabiliser Chain

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PermutationsIntroduction to GroupsAlice NiemeyerThe University ofWestern AustraliaΩ finite set. A permutation of Ω is a 1-1and onto function from Ω to Ω.The image of b ∈ Ω under a permutation gis denoted b g .GroupsHexagon SymmetriesPermutationsPermutation GroupsGeneratorsOrbitsCosetsStabilisersOrbit Stabiliser TheoremBase & Stabiliser Chain

¡ ¡¡ ¡¡ ¡¨¨¨¨¨¨Disjoint Cycle NotationIntroduction to GroupsAlice NiemeyerThe University ofWestern AustraliaConsider again the reflection through theline passing through 1 and 4 of the hexagon.1( )£ ¢ £ ¢£ £ ¢ ¢£ £ ¢ ¢11 2 3 4 5 66 1 6 5 4 3 22621234GroupsHexagon SymmetriesPermutationsPermutation GroupsGeneratorsOrbitsCosetsStabilisersOrbit Stabiliser TheoremBase & Stabiliser Chain35 3¤ ¤ ¥ ¥¦ § § ¦¥ ¤ ¦ ¤¦ § §¥§ § ¦ ¦ ¥ ¥ ¤ ¤© ©© ©© ©4456(1)(2, 6)(3, 5)(4)5

ConventionsIntroduction to GroupsAlice NiemeyerThe University ofWestern AustraliaI () denotes the identity permutation.I g k denotes g · g · · · g .} {{ }kI Cycles with just one point are omitted.GroupsHexagon SymmetriesPermutationsPermutation GroupsGeneratorsOrbitsCosetsStabilisersOrbit Stabiliser TheoremBase & Stabiliser ChainExample(1)(2, 6)(3, 5)(4) is written as (2, 6)(3, 5).

Order of a PermutationIntroduction to GroupsAlice NiemeyerThe University ofWestern AustraliaLet g be a permutation on n points.The order of g is the smallest m such thatg m = (). Notation o(g).GroupsHexagon SymmetriesPermutationsPermutation GroupsGeneratorsOrbitsCosetsStabilisersOrbit Stabiliser TheoremBase & Stabiliser Chain

Order of a PermutationIntroduction to GroupsAlice NiemeyerThe University ofWestern AustraliaTheorem (Ruffini 1799)The order of a permutation in disjoint cyclenotation is the least common multiple of thecycle lengths.GroupsHexagon SymmetriesPermutationsPermutation GroupsGeneratorsOrbitsCosetsStabilisersOrbit Stabiliser TheoremBase & Stabiliser Chain

Order of a PermutationIntroduction to GroupsAlice NiemeyerThe University ofWestern AustraliaTheorem (Ruffini 1799)The order of a permutation in disjoint cyclenotation is the least common multiple of thecycle lengths.Exampleo((2, 6)(3, 5)) = lcm(2, 2) = 2.GroupsHexagon SymmetriesPermutationsPermutation GroupsGeneratorsOrbitsCosetsStabilisersOrbit Stabiliser TheoremBase & Stabiliser Chain

Cycle StructureExample for Ω = {1, . . . , 12}:g = (1, 3, 4)(2, 5, 7)(6, 8, 11, 10)(9, 12)has one cycle of length 2, two cycles oflength 3 and one cycle of length 4.Introduction to GroupsAlice NiemeyerThe University ofWestern AustraliaGroupsHexagon SymmetriesPermutationsPermutation GroupsGeneratorsOrbitsCosetsStabilisersOrbit Stabiliser TheoremBase & Stabiliser ChainCyc(g) = 1 0 2 1 3 2 4 1 5 0 · · · 12 0 .

Cycle StructureIntroduction to GroupsAlice NiemeyerThe University ofWestern AustraliaIf g has α i cycles of length i we say g hascycle structure1 α 12 α2 · · · n α n.GroupsHexagon SymmetriesPermutationsPermutation GroupsGeneratorsOrbitsCosetsStabilisersOrbit Stabiliser TheoremBase & Stabiliser Chain

Conjugate PermutationsLet g and h be permutations on Ω. Thenh −1 gh is the conjugate of g by h.Notation: g h .If a g = b then(a h ) h−1 gh= b h .Introduction to GroupsAlice NiemeyerThe University ofWestern AustraliaGroupsHexagon SymmetriesPermutationsPermutation GroupsGeneratorsOrbitsCosetsStabilisersOrbit Stabiliser TheoremBase & Stabiliser Chain

Conjugate PermutationsLet g and h be permutations on Ω. Thenh −1 gh is the conjugate of g by h.Notation: g h .a 6a 1ga 2ha 6ha 1g hha 2Introduction to GroupsAlice NiemeyerThe University ofWestern AustraliaGroupsHexagon SymmetriesPermutationsPermutation GroupsGeneratorsOrbitsCosetsStabilisersOrbit Stabiliser TheoremBase & Stabiliser Chaina 5a 3ha 5ha 3a 4 a 4h

Conjugate PermutationsLet g and h be permutations on Ω. Thenh −1 gh is the conjugate of g by h.Notation: g h .LemmaConjugate permutations have the same cyclestructure.Introduction to GroupsAlice NiemeyerThe University ofWestern AustraliaGroupsHexagon SymmetriesPermutationsPermutation GroupsGeneratorsOrbitsCosetsStabilisersOrbit Stabiliser TheoremBase & Stabiliser Chain

Conjugate PermutationsLet g and h be permutations on Ω. Thenh −1 gh is the conjugate of g by h.Notation: g h .Example(1, 2, 5)(7, 3, 6) (1,7)(2,6)= (7, 6, 5)(1, 3, 2)= (5, 7, 6)(1, 3, 2).Introduction to GroupsAlice NiemeyerThe University ofWestern AustraliaGroupsHexagon SymmetriesPermutationsPermutation GroupsGeneratorsOrbitsCosetsStabilisersOrbit Stabiliser TheoremBase & Stabiliser Chain

Permutation GroupsIntroduction to GroupsAlice NiemeyerThe University ofWestern AustraliaTheoremThe permutations of Ω form a group underfunction composition.Symmetric GroupsThe group of all permutations on Ω is calledthe Symmetric Group on Ω, denotedSym(Ω) or S n if Ω = {1, . . . , n}.GroupsHexagon SymmetriesPermutationsPermutation GroupsGeneratorsOrbitsCosetsStabilisersOrbit Stabiliser TheoremBase & Stabiliser Chain

Permutation Groupsarise in many contexts:I Group TheoryI Graph TheoryI CombinatoricsI Design TheoryI CryptographyI Symmetry and SearchI etc. etc.Introduction to GroupsAlice NiemeyerThe University ofWestern AustraliaGroupsHexagon SymmetriesPermutationsPermutation GroupsGeneratorsOrbitsCosetsStabilisersOrbit Stabiliser TheoremBase & Stabiliser Chain

QuestionsIntroduction to GroupsAlice NiemeyerThe University ofWestern AustraliaGiven G a permutation group on Ω.1. How many elements does G have?(order of G)2. Is a given permutation of Ω in G?3. Is G isomorphic to a known group?4. What are the subgroups of G?5. And many more.GroupsHexagon SymmetriesPermutationsPermutation GroupsGeneratorsOrbitsCosetsStabilisersOrbit Stabiliser TheoremBase & Stabiliser Chain

Symmetries of n-gonsHexagonThe symmetries of the hexagon form a groupcalled Dihedral Group of order 12, denotedD 12 . It has 12 elements.n-gonThe symmetries of the n-gon form a groupcalled Dihedral Group of order 2n, denotedD 2n . It has 2n elements.Introduction to GroupsAlice NiemeyerThe University ofWestern AustraliaGroupsHexagon SymmetriesPermutationsPermutation GroupsGeneratorsOrbitsCosetsStabilisersOrbit Stabiliser TheoremBase & Stabiliser Chain

GeneratorsIntroduction to GroupsAlice NiemeyerThe University ofWestern AustraliaGiven some permutations on Ω the smallestgroup of all permutations on Ω containingthe given permutations is called the groupgenerated by the given permutations.GroupsHexagon SymmetriesPermutationsPermutation GroupsGeneratorsOrbitsCosetsStabilisersOrbit Stabiliser TheoremBase & Stabiliser Chain

Generators for D 12Introduction to GroupsAlice NiemeyerThe University ofWestern AustraliaConsider g = (1, 2, 3, 4, 5, 6) andh = (2, 6)(3, 5). The group generated by gand h is denoted 〈g, h〉 and is D 12 .For example,〈g, h〉 has to containg · h = (1, 6)(2, 5)(3, 4),g · g · h = (1, 5)(2, 4).GroupsHexagon SymmetriesPermutationsPermutation GroupsGeneratorsOrbitsCosetsStabilisersOrbit Stabiliser TheoremBase & Stabiliser Chain

Introduction to GroupsAlice NiemeyerThe University ofWestern AustraliaD 12 = {(), (2, 6)(3, 5), (1, 2)(3, 6)(4, 5),(1, 2, 3, 4, 5, 6), (1, 3)(4, 6),(1, 3, 5)(2, 4, 6), (1, 4)(2, 3)(5, 6),(1, 4)(2, 5)(3, 6), (1, 5)(2, 4),(1, 5, 3)(2, 6, 4), (1, 6, 5, 4, 3, 2),(1, 6)(2, 5)(3, 4)}GroupsHexagon SymmetriesPermutationsPermutation GroupsGeneratorsOrbitsCosetsStabilisersOrbit Stabiliser TheoremBase & Stabiliser Chain

OrbitIntroduction to GroupsAlice NiemeyerThe University ofWestern AustraliaLet a ∈ Ω. The orbit of a under G isorb G (a) = {a g | g ∈ G}.GroupsHexagon SymmetriesPermutationsPermutation GroupsGeneratorsOrbitsCosetsStabilisersOrbit Stabiliser TheoremBase & Stabiliser Chain

QuestionsIntroduction to GroupsAlice NiemeyerThe University ofWestern AustraliaI How can we compute the orbit of apoint under a group?I How much do we need to know aboutthe group?GroupsHexagon SymmetriesPermutationsPermutation GroupsGeneratorsOrbitsCosetsStabilisersOrbit Stabiliser TheoremBase & Stabiliser Chain

Computing OrbitsIt is enough to know generators. Forexample,D 12 = 〈g = (1, 2, 3, 4, 5, 6), h = (2, 6)(3, 5)〉.k ∈ D 12 is of the form g · g · h · · · g · h.If 1 k = j thenj = ((((1 g ) g ) h )···g ) h .Introduction to GroupsAlice NiemeyerThe University ofWestern AustraliaGroupsHexagon SymmetriesPermutationsPermutation GroupsGeneratorsOrbitsCosetsStabilisersOrbit Stabiliser TheoremBase & Stabiliser ChainHence we only need to see what g and h doto points in the orbit.

Computing OrbitsIt is enough to know generators. Forexample,D 12 = 〈g = (1, 2, 3, 4, 5, 6), h = (2, 6)(3, 5)〉.1hgg2hghg6hh53gg4Introduction to GroupsAlice NiemeyerThe University ofWestern AustraliaGroupsHexagon SymmetriesPermutationsPermutation GroupsGeneratorsOrbitsCosetsStabilisersOrbit Stabiliser TheoremBase & Stabiliser Chainh

Computing OrbitsIt is enough to know generators. Forexample,D 12 = 〈g = (1, 2, 3, 4, 5, 6), h = (2, 6)(3, 5)〉.orb D12 (1) = {1, 2, 3, 4, 5, 6}.Introduction to GroupsAlice NiemeyerThe University ofWestern AustraliaGroupsHexagon SymmetriesPermutationsPermutation GroupsGeneratorsOrbitsCosetsStabilisersOrbit Stabiliser TheoremBase & Stabiliser Chain

ExampleIntroduction to GroupsAlice NiemeyerThe University ofWestern AustraliaTest the Orbit-Algorithm on PSL 2 (13)G = 〈(3, 13, 11, 9, 7, 5)(4, 14, 12, 10, 8, 6),(1, 2, 9)(3, 8, 10)(4, 5, 12)(6, 13, 14)〉GroupsHexagon SymmetriesPermutationsPermutation GroupsGeneratorsOrbitsCosetsStabilisersOrbit Stabiliser TheoremBase & Stabiliser ChainWhat is the orbit containing 1?

Introduction to GroupsAlice NiemeyerThe University ofWestern AustraliaDefinitionG is called transitive if orb G (a) = Ω fora ∈ Ω.GroupsHexagon SymmetriesPermutationsPermutation GroupsGeneratorsOrbitsCosetsStabilisersOrbit Stabiliser TheoremBase & Stabiliser Chain

CosetsIntroduction to GroupsAlice NiemeyerThe University ofWestern AustraliaIf G is a group and H a subgroup of G thenthe cosets of H in G are the setsHg for g ∈ G.GroupsHexagon SymmetriesPermutationsPermutation GroupsGeneratorsOrbitsCosetsStabilisersOrbit Stabiliser TheoremBase & Stabiliser ChainThe cosets partition G.

Why Cosets?Introduction to GroupsAlice NiemeyerThe University ofWestern AustraliaThe elements in the coset Hg of H in G canbe thought of as being equivalent modulo H,that is, they differ only by an element of H.The information provided by H is ignored.GroupsHexagon SymmetriesPermutationsPermutation GroupsGeneratorsOrbitsCosetsStabilisersOrbit Stabiliser TheoremBase & Stabiliser Chain

ExampleIntroduction to GroupsAlice NiemeyerThe University ofWestern AustraliaThink of competing sports teams.I Which team won?I Not: which player won?I Competition time table:When is team A playing team B?Not: When is player X playing againstplayer Y?GroupsHexagon SymmetriesPermutationsPermutation GroupsGeneratorsOrbitsCosetsStabilisersOrbit Stabiliser TheoremBase & Stabiliser Chain

Introduction to GroupsAlice NiemeyerThe University ofWestern AustraliaLagrange’s TheoremThe number [G : H] of cosets of H in Gdivides |G| and[G : H] = |G||H| .GroupsHexagon SymmetriesPermutationsPermutation GroupsGeneratorsOrbitsCosetsStabilisersOrbit Stabiliser TheoremBase & Stabiliser Chain

ExampleH = 〈(1, 2, 3, 4, 5, 6)〉 ≤ D 12 . The cosets ofH in G areH = { (), (1, 2, 3, 4, 5, 6), (1, 3, 5)(2, 4, 6),(1, 4)(2, 5)(3, 6), (1, 5, 3)(2, 6, 4),(1, 6, 5, 4, 3, 2)} andH ∗ (2, 6)(3, 5) = {(2, 6)(3, 5),(1, 6)(2, 5)(3, 4), (1, 5)(2, 4),(1, 4)(2, 3)(5, 6), (1, 3)(4, 6),(1, 2)(3, 6)(4, 5)}Introduction to GroupsAlice NiemeyerThe University ofWestern AustraliaGroupsHexagon SymmetriesPermutationsPermutation GroupsGeneratorsOrbitsCosetsStabilisersOrbit Stabiliser TheoremBase & Stabiliser Chain

StabiliserIntroduction to GroupsAlice NiemeyerThe University ofWestern AustraliaLet a ∈ Ω. The Stabiliser of a under G isStab G (a) = {g ∈ G | a g = a}.GroupsHexagon SymmetriesPermutationsPermutation GroupsGeneratorsOrbitsCosetsStabilisersOrbit Stabiliser TheoremBase & Stabiliser ChainStab G (a) is a subgroup of G (a subset whichis a group).

Stabiliser of 1 in D 12Introduction to GroupsAlice NiemeyerThe University ofWestern AustraliaWhich permutations of the hexagon fix 1?GroupsHexagon SymmetriesPermutationsPermutation GroupsGeneratorsOrbitsCosetsStabilisersOrbit Stabiliser TheoremBase & Stabiliser Chain

Stabiliser of 1 in D 12Introduction to GroupsAlice NiemeyerThe University ofWestern AustraliaWhich permutations of the hexagon fix 1?No rotation fixes 1.Only () and the reflection along the linethrough 1 and 4 fixes 1.GroupsHexagon SymmetriesPermutationsPermutation GroupsGeneratorsOrbitsCosetsStabilisersOrbit Stabiliser TheoremBase & Stabiliser ChainStab D12 (1) = {(), (2, 6)(3, 5)}.

Orbit-Stabiliser TheoremIntroduction to GroupsAlice NiemeyerThe University ofWestern AustraliaTheoremLet G be a permutation group on Ω. Thenfor a ∈ Ω[G : Stab G (a)] = |orb G (a)|,GroupsHexagon SymmetriesPermutationsPermutation GroupsGeneratorsOrbitsCosetsStabilisersOrbit Stabiliser TheoremBase & Stabiliser Chainthat is the number of cosets of Stab G (a) inG equals the number of points in the orbitof a under G.

Orbit-Stabiliser TheoremIntroduction to GroupsAlice NiemeyerThe University ofWestern AustraliaI All h ∈ Stab G (a) map a to a.I All h ∈ Stab G (a)g map a to a g .CosetsAll elements in a coset of Stab G (a) map a tothe same point.GroupsHexagon SymmetriesPermutationsPermutation GroupsGeneratorsOrbitsCosetsStabilisersOrbit Stabiliser TheoremBase & Stabiliser Chain

Repeated StabilisersIntroduction to GroupsAlice NiemeyerThe University ofWestern AustraliaLet G be a permutation group. Choosea ∈ Ω and let G (2) = Stab G (a). Then G (2) isagain a group and we can choose anotherb ∈ Ω and compute G (3) = Stab G (2)(b).How often can we repeat this?GroupsHexagon SymmetriesPermutationsPermutation GroupsGeneratorsOrbitsCosetsStabilisersOrbit Stabiliser TheoremBase & Stabiliser Chain

BaseLet B be a sequence of points[a 1 , a 2 , . . . , a k ]. Define= GG (2) = Stab G(1)(a 1 )G (1).G (i+1) = Stab G(i)(a i )Introduction to GroupsAlice NiemeyerThe University ofWestern AustraliaGroupsHexagon SymmetriesPermutationsPermutation GroupsGeneratorsOrbitsCosetsStabilisersOrbit Stabiliser TheoremBase & Stabiliser ChainBaseB is a base if G (k+1) = {1}.

Introduction to GroupsAlice NiemeyerThe University ofWestern AustraliaStabiliser ChainG = G (1) ≥ G (2) ≥ · · · G (k+1) = {1}.A base is irredundant if the inclusions areproper.GroupsHexagon SymmetriesPermutationsPermutation GroupsGeneratorsOrbitsCosetsStabilisersOrbit Stabiliser TheoremBase & Stabiliser Chain

Order of a Permutation GroupIntroduction to GroupsAlice NiemeyerThe University ofWestern AustraliaBy the orbit stabiliser we haveOrder of G|G| = [G (1) : G (2) ] · · · [G (k) : G (k+1) ]= |orb G(1)(a 1 )||orb G(2)(a 2 )| · · · |orb G(k)(a k )|GroupsHexagon SymmetriesPermutationsPermutation GroupsGeneratorsOrbitsCosetsStabilisersOrbit Stabiliser TheoremBase & Stabiliser Chain

Base and Stabiliser ChainIntroduction to GroupsAlice NiemeyerThe University ofWestern AustraliaHence knowing a base and a stabiliser chainfor a group allows us to compute the orderof the group.GroupsHexagon SymmetriesPermutationsPermutation GroupsGeneratorsOrbitsCosetsStabilisersOrbit Stabiliser TheoremBase & Stabiliser Chain

Example G = D 12Let G (1) = G.Choose a 1 = 1. Then we knowG (2) = Stab G(1)(a 1 ) = {(), (2, 6)(3, 5)}.Let a 2 = 2. Then the only element of G (2)that fixes 2 is (). HenceIntroduction to GroupsAlice NiemeyerThe University ofWestern AustraliaGroupsHexagon SymmetriesPermutationsPermutation GroupsGeneratorsOrbitsCosetsStabilisersOrbit Stabiliser TheoremBase & Stabiliser ChainG (3) = Stab G(2)(a 2 ) = {()}.

Example G = D 12Introduction to GroupsAlice NiemeyerThe University ofWestern AustraliaWe get the base [1, 2] and the stabiliserchain:G = G (1) > G (2) > G (3) = {1}.GroupsHexagon SymmetriesPermutationsPermutation GroupsGeneratorsOrbitsCosetsStabilisersOrbit Stabiliser TheoremBase & Stabiliser Chain

Example G = D 12Introduction to GroupsAlice NiemeyerThe University ofWestern AustraliaNow we use the Orbit-Stabiliser Theorem toverify |G| = 12.|G| = |orb G(1)(a 1 )||orb G(2)(a 2 )|.Now orb G (1)(1) = {1, 2, 3, 4, 5, 6}and orb G (2)(2) = {2, 6}GroupsHexagon SymmetriesPermutationsPermutation GroupsGeneratorsOrbitsCosetsStabilisersOrbit Stabiliser TheoremBase & Stabiliser Chain|G| = 6 · 2 = 12.

Why?Introduction to GroupsAlice NiemeyerThe University ofWestern AustraliaWe use this method often.For example, if we know H ⊆ G and wewant to prove H = G.We compute a few stabilisers in the stabiliserchains until we reach small enough groups.If the orbit lengths of the correspondingstabilisers in G and H were equal and thefinal groups are equal, then G = H.GroupsHexagon SymmetriesPermutationsPermutation GroupsGeneratorsOrbitsCosetsStabilisersOrbit Stabiliser TheoremBase & Stabiliser Chain

When?Introduction to GroupsAlice NiemeyerThe University ofWestern AustraliaWe use this method, for example, to provethat the full group of symmetries of a givenobject is a particular group.GroupsHexagon SymmetriesPermutationsPermutation GroupsGeneratorsOrbitsCosetsStabilisersOrbit Stabiliser TheoremBase & Stabiliser Chain

Schreier-Sims AlgorithmIntroduction to GroupsAlice NiemeyerThe University ofWestern AustraliaMany questions about finite permutationgroups are answered by the Schreier-SimsAlgorithm.It computes a base for a group and a set ofgenerators for each stabiliser group in thechain.GroupsHexagon SymmetriesPermutationsPermutation GroupsGeneratorsOrbitsCosetsStabilisersOrbit Stabiliser TheoremBase & Stabiliser Chain

Large Base GroupsIntroduction to GroupsAlice NiemeyerThe University ofWestern AustraliaLet G be an infinite family of permutationgroups.G is small-base if each G ∈ G has a basewith at most b log c (deg(G)) elements,where b and c are positive constants.Otherwise G is called large base.GroupsHexagon SymmetriesPermutationsPermutation GroupsGeneratorsOrbitsCosetsStabilisersOrbit Stabiliser TheoremBase & Stabiliser Chain

ExampleIntroduction to GroupsAlice NiemeyerThe University ofWestern Australia{S n | n ∈ N} is a family of large base groups.GroupsHexagon SymmetriesPermutationsPermutation GroupsGeneratorsOrbitsCosetsStabilisersOrbit Stabiliser TheoremBase & Stabiliser Chain

Small Base GroupsIntroduction to GroupsAlice NiemeyerThe University ofWestern AustraliaI can be handled efficiently withSchreier-Sims algorithmGroupsHexagon SymmetriesPermutationsPermutation GroupsGeneratorsOrbitsCosetsStabilisersOrbit Stabiliser TheoremBase & Stabiliser Chain

Large Base GroupsIntroduction to GroupsAlice NiemeyerThe University ofWestern AustraliaI require specialised randomisedalgorithms.I We’ll consider an example in the lasttalk.GroupsHexagon SymmetriesPermutationsPermutation GroupsGeneratorsOrbitsCosetsStabilisersOrbit Stabiliser TheoremBase & Stabiliser Chain