# Alexander Hulpke Department of Mathematics Colorado State University Fort Collins, CO, 80523, USA

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1 Finding Subgroups Alexander Hulpke Department of Mathematics Colorado State University Fort Collins, CO, 80523, USA I

2 A Popular Problem Given a finite (permutation, matrix) group G, find subgroups: Frequent task (CANNON, 2011) -Not all, but specific properties, one or all. -Up to conjugacy U ~ g -1 Ug. There is no universal algorithm (how to specify desired properties). Best strategies: - Search in smaller groups (sub- or factor-) - Use hooks in existing algorithms (or put in own hooks). Thus survey what exists.

3 Admission of Failure How to organize this? One obviously would like an organization by task. I only can give a classification by algorithm (showing you a nice wrench set instead of repairing your car). Still, I hope this might help to give an idea where to put limitation hooks (and what kinds of hooks are suitable).

4 Possible Strategies Build Up Just Find It Between Build Down

5 Direct Constructions Direct methods for the construction of particular subgroups without the need to build part of the subgroup lattice.

6 Direct Constructions Cyclic Subgroups Direct methods for the construction of particular subgroups without the need to build part of the subgroup lattice. Stabilizers Sylow Subgroups PCore Composition Series, Radical

7 Cyclic Subgroups Cyclic Subgroups There are algorithms that compute conjugacy class representatives of elements. Eliminate coprime powers g x, gcd( g,x)=1, to get cyclic Stabilizers Sylow Subgroups PCore Composition Series, Radical subgroups up to conjugacy.

8 Cyclic Subgroups There are algorithms that compute conjugacy class representatives of elements. Eliminate coprime powers g x, gcd( g,x)=1, to get cyclic subgroups up to conjugacy. NEUBÜSER, MECKY, 1989 CANNON, HOLT, 2006 Cyclic Subgroups Stabilizers Sylow Subgroups PCore CANNON, SOUVIGNIER 1997 H. Composition 2000, Series, Radical 2013

9 Cyclic Subgroups Cyclic Subgroups There are algorithms that compute conjugacy class representatives of elements. Eliminate coprime powers g x, gcd( g,x)=1, to get cyclic Stabilizers Sylow Subgroups PCore Composition Series, Radical subgroups up to conjugacy.

10 Orbit / Stabilizer If G acts on Ω, the Orbit/ Stabilizer algorithm finds the orbit of ω Ω by repeated computation of images under generators. Schreier generators for StabG(ω). Cyclic Subgroups Stabilizers Sylow Subgroups PCore Composition Series, Radical

11 Orbit / Stabilizer If G acts on Ω, the Orbit/ Stabilizer algorithm finds the orbit of ω Ω by repeated computation of images under generators. Schreier generators for StabG(ω). Problems: Cyclic Subgroups Stabilizers Sylow Subgroups PCore Composition Series, Radical Cost (time and memory) is proportional to stabilizer index. Not every subgroup is a natural stabilizer.

12 Backtrack Backtrack search algorithms for particular actions of perm. groups: (Point, Tuple), Set stabilizer Element Centralizer Subgroup Normalizer Intersection Cyclic Subgroups Stabilizers Sylow Subgroups PCore Composition Series, Radical

13 Backtrack Backtrack search algorithms for particular actions of perm. groups: (Point, Tuple), Set stabilizer Element Centralizer Subgroup Normalizer Intersection Cyclic Subgroups Stabilizers Sylow Subgroups PCore Composition Series, Radical Problems: Inherently exponential time, hard to analyze. Very hard to extend.

14 Sylow Structure Sylow subgroups have a nontrivial centre. They thus lie in the centralizer of an element of order p. Compute these centralizers and then search in the centre Cyclic Subgroups Stabilizers Sylow Subgroups PCore Composition Series, Radical factor. PCore as intersection of conjugates.

15 Composition Structure Composition series by finding kernels of nontrivial homomorphisms. (Transitive, Imprimitive actions; them analysis of primitive case.) (Solvable) Radical is the kernel Cyclic Subgroups Stabilizers Sylow Subgroups PCore Composition Series, Radical of the action of G on nonabelian composition factors.

16 Build Up It seems plausible to try to build up subgroups from smaller ones, starting with cyclic subgroups.

17 Build Up It seems plausible to try to build up subgroups from Find Generators Cyclic Extension smaller ones, starting with cyclic subgroups. Indeed this is possible. A main difficulty is that it will generate lots of irrelevant and redundant groups and conjugacy tests for subgroups are expensive.

18 Find Generators Any U<G is generated by a set of elements u1,...,uk. Each ui lies in conj. class. Thus classify such tuples (with particular properties, e.g. presentation) up to conjugacy (CG(u) orbits on x G are given by double cosets CG(x)\G/CG(u). GAP-Function: IsomorphicSubgroups, TomDataSubgroupsAlmostSimple Hook: Internal function MorClassLoop

19 Find Generators Any U<G is generated by a set of elements u1,...,uk. Each ui lies in conj. class. Thus classify such tuples (with particular properties, e.g. presentation) up to conjugacy (CG(u) orbits on x G are given by double cosets CG(x)\G/CG(u). GAP-Function: IsomorphicSubgroups, TomDataSubgroupsAlmostSimple Hook: Internal function MorClassLoop

20 Cyclic Extension More systematically, construct U as U= V,x with smaller V, x NG(V). Seed with perfect subgroups. Find via generator images from catalog. GAP-Function: LatticeByCyclicExtension Hook: Function to discard V, flag to skip nontrivial perfect subgroups. Problems: Many, expensive conjugacy tests. Store by list of cyclic subgroups.

21 Cyclic Extension More systematically, construct U as U= V,x with smaller V, x NG(V). Seed with perfect subgroups. Find via generator images from catalog. GAP-Function: LatticeByCyclicExtension Hook: Function to discard V, flag to skip nontrivial perfect subgroups. NEUBÜSER 1960 Problems: Many, expensive conjugacy tests. Store by list of cyclic subgroups.

22 Cyclic Extension More systematically, construct U as U= V,x with smaller V, x NG(V). Seed with perfect subgroups. Find via generator images from catalog. GAP-Function: LatticeByCyclicExtension Hook: Function to discard V, flag to skip nontrivial perfect subgroups. Problems: Many, expensive conjugacy tests. Store by list of cyclic subgroups.

23 Cyclic Extension More systematically, construct U as U= V,x with smaller V, x NG(V). Seed with perfect subgroups. Find via generator images from catalog. GAP-Function: LatticeByCyclicExtension Hook: Function to discard V, flag to skip nontrivial perfect subgroups. Problems: Many, expensive conjugacy tests. Store by list of cyclic subgroups.

24 Complements G Intermezzo For N G, a complement is a N C C2 C3 subgroup C G such that N C= 1 and NC=G. 1 Complements fall in classes under N. general there can be multiple (or no) classes. Sufficient to find representatives. As C G/N, consider as homomorphisms G G with kernel N. Every g G is mapped to g ng with the cofactor ng N. Find these ng for generators of G/N.

25 Complements G For N G, a complement is a N C C2 C3 subgroup C G such that N C= 1 and NC=G. 1 Complements fall in classes under N. In general there can be multiple (or no) classes. Sufficient to find representatives. As C G/N, consider as homomorphisms G G with kernel N. Every g G is mapped to g ng with the cofactor ng N. Find these ng for generators of G/N.

26 Technical DETAILS Linear Algebra for Cocycles If N is elementary abelian then one can collect these equations into a G/N-part and an N-part. E.g. a relator Na Nb Na for G/N becomes a na b nb a na = a b na b nb a na = a b a na ba nb a na Thus in G, (a b a) 1 N equals the (linearly written!) vector space element na (Mba+1)+ nb Ma where Mx is the action induced by x on N. For given Mx, the ny are variables. These equations for all relators yield a linear inhomogeneous system of equations, solutions are cocycles Z 1. Conjugacy by N gives B 1.

27 Technical DETAILS Linear Algebra for Cocycles If N is elementary abelian then one can collect these equations into a G/N-part and an N-part. E.g. a relator Na Nb Na for G/N becomes a na b nb a na = a b na b nb a na = a b a na ba nb a na Thus in G, (a b a) 1 N equals the (linearly written!) vector space element na (Mba+1)+ nb Ma where Mx is the action induced by x on N. For given Mx, the ny are variables. These equations for all relators yield a linear inhomogeneous system of equations, solutions are cocycles Z 1. Conjugacy by N gives B 1. CELLER, NEUBÜSER, WRIGHT 1990

28 Technical DETAILS Linear Algebra for Cocycles If N is elementary abelian then one can collect these equations into a G/N-part and an N-part. E.g. a relator Na Nb Na for G/N becomes a na b nb a na = a b na b nb a na = a b a na ba nb a na Thus in G, (a b a) 1 N equals the (linearly written!) vector space element na (Mba+1)+ nb Ma where Mx is the action induced by x on N. For given Mx, the ny are variables. These equations for all relators yield a linear inhomogeneous system of equations, solutions are cocycles Z 1. Conjugacy by N gives B 1.

29 Technical DETAILS Solvable Normal Subgroup If N is solvable, take a G-normal series for N with elementary abelian factors. In each step take M G elementary abelian with N M. Assume we found representatives Ci for the complements to N/M in G/M. Then for each Ci find representatives for the complements to M in Ci and fuse these under action of NG(Ci). N G M C... 1 C i C k...

30 Technical DETAILS Solvable Normal Subgroup If N is solvable, take a G-normal series for N with elementary As G/N=C, elements abelian factors. of G induce no outer automorphisms of Ci In each step take M G elementary abelian with N M. Assume we found representatives Ci for the complements to N/M in G/M. Then for each Ci find representatives for the complements to M in Ci and fuse these under action of XXXX NG(Ci). CG(Ci) G N C... 1 C i C k M...

31 Technical DETAILS Solvable factor If G/N is solvable, there is A G such that A/N G/N is a normal p-subgroup. Any complement C in G contains a complement D C in A where D is a p-subgroup. Up to conjugacy D S, where S A is a fixed p-sylow subgroup. As A G we have that D C, and thus C NG(D)=:B. A B=NG(D) S N G S N D 1 C

32 In Between For a given U<G the subgroups U<V<G correspond to block systems for the action of G on cosets of U.

33 In Between For a given U<G the subgroups U<V<G correspond to block systems for the action of G on cosets of U.

34 In Between For a given U<G the subgroups U<V<G correspond to block systems for the action of G on cosets of U.

35 In Between For a given U<G the subgroups U<V<G correspond to block systems for the action of G on cosets of U. Finding blocks is easy: Take a potential block seed containing two points (orbit of point stabilizer). Form Images, combine to partition (fusing cells if neccessary).

36 In Between For a given U<G the subgroups U<V<G correspond to block systems for the action of G on cosets of U. Finding blocks is easy: Take a potential block seed containing two points (orbit of point stabilizer). Form Images, combine to partition (fusing cells if neccessary).

37 In Between For a given U<G the subgroups U<V<G correspond to block systems for the action of G on cosets of U. Finding blocks is easy: Take a potential block seed containing two points (orbit of point stabilizer). Form Images, combine to partition (fusing cells if neccessary).

38 In Between For a given U<G the subgroups U<V<G correspond to block systems for the action of G on cosets of U. Finding blocks is easy: Take a potential block seed containing two points (orbit of point stabilizer). Form Images, combine to partition (fusing cells if neccessary).

39 In Between For a given U<G the subgroups U<V<G correspond to block systems for the action of G on cosets of U. Finding blocks is easy: Take a potential block seed containing two points (orbit of point stabilizer). Form Images, combine to partition (fusing cells if neccessary).

40 In Between For a given U<G the subgroups U<V<G correspond to block systems for the action of G on cosets of U. Finding blocks is easy: Take a potential block seed containing two points (orbit of point stabilizer). Form Images, combine to partition (fusing cells if neccessary).

41 In Between For a given U<G the subgroups U<V<G correspond to block systems for the action of G on cosets of U. Finding blocks is easy: Take a potential block seed containing two points (orbit of point stabilizer). Form Images, combine to partition (fusing cells if neccessary).

42 In Between For a given U<G the subgroups U<V<G correspond to block systems for the action of G on cosets of U. Finding blocks is easy: Take a potential block seed containing two points (orbit of point stabilizer). Form Images, combine to partition (fusing cells if neccessary).

43 In Between For a given U<G the subgroups U<V<G correspond to block systems for the action of G on cosets of U. Subgroups Between Maximal Subgroups Maximality Test (Future thesis topic... ) Finding blocks is easy: Take a potential block seed containing two points (orbit of point stabilizer). Form Images, combine to partition (fusing cells if neccessary).

44 Subgroups Between Take the permutation action on subgroups and construct all block systems systematically. This yields all subgroups between two. GAP-Function: IntermediateSubgroups, AscendingChain. Caveat: Needs to write down permutations for action on cosets. Inefficient if there are many blocks.

45 Primitive Groups If a permutation group G affords no nontrivial block systems, it is primitive. The O NAN-SCOTT Theorem states that then Soc(G)=T n, T simple and either Soc(G)=p n and G AGLn(q). StabG(1) complements Soc(G)=p n. Or Soc(G)=T n nonabelian, G Aut(T) Sn. StabG(1) StabSoc(1) which combines diagonal/direct product of maximals.

46 Primitive Groups If a permutation group G affords no nontrivial block systems, it is primitive. The O NAN-SCOTT Theorem states that then Soc(G)=T n, T simple and either Soc(G)=p n and G AGLn(q). StabG(1) complements Soc(G)=p n. Or Soc(G)=T n nonabelian, G Aut(T) Sn. Using Cohomology StabG(1) StabSoc(1) which combines diagonal/direct product of maximals.

47 Primitive Groups If a permutation group G affords no nontrivial block systems, it is primitive. The O NAN-SCOTT Theorem states that then Soc(G)=T n, T simple and either Soc(G)=p n and G AGLn(q). StabG(1) complements Soc(G)=p n. Or Soc(G)=T n nonabelian, G Aut(T) Sn. StabG(1) StabSoc(1) which combines diagonal/direct product of maximals.

48 Primitive Groups If a permutation group G affords no {(t,t,...,t ) t T} nontrivial block systems, it is primitive. The O NAN-SCOTT Theorem states that then Soc(G)=T n, T simple and either Soc(G)=p n and G AGLn(q). StabG(1) complements Soc(G)=p n. Or Soc(G)=T n nonabelian, G Aut(T) Sn. StabG(1) StabSoc(1) which combines diagonal/direct product of maximals.

49 Primitive Groups If a permutation group G affords no {(t,t,...,t ) t T} nontrivial block systems, {(m1,m2,...,mn) it is primitive. mi M } The O NAN-SCOTT Theorem M maximal states in T. that then Soc(G)=T n, T simple and either Soc(G)=p n and G AGLn(q). StabG(1) complements Soc(G)=p n. Or Soc(G)=T n nonabelian, G Aut(T) Sn. StabG(1) StabSoc(1) which combines diagonal/direct product of maximals.

50 Primitive Groups If a permutation group G affords no {(t,t,...,t ) t T} nontrivial block systems, {(m1,m2,...,mn) it is primitive. mi M } The O NAN-SCOTT Theorem M maximal states in T. that then Soc(G)=T n, T simple and either Soc(G)=p {(m1,m1,...,m1,...,mk,mk,...,mk) n and G AGLn(q). mi M StabG(1) }, n=kl complements Soc(G)=p n. Or Soc(G)=T n nonabelian, G Aut(T) Sn. StabG(1) StabSoc(1) which combines diagonal/direct product of maximals.

51 Find Maximal Subgroups The possible socles arise from chief factors M/N of G. Homomorphism φ:g action on M/N. In the Image φ construct maximal subgroups as normalizers of these diagonals and direct products. Base case: M/N simple, maxes of simple. For the abelian case need complements to abelian chief factors.

52 Find Maximal Subgroups The possible socles arise from chief factors M/N of G. Homomorphism φ:g action on M/N. In the Image φ construct maximal subgroups as normalizers of these diagonals and direct products. Base case: M/N simple, maxes of simple. For the abelian case need complements to abelian chief factors. CANNON, HOLT 2004 EICK, H. 2001

53 Find Maximal Subgroups The possible socles arise from chief factors M/N of G. Homomorphism φ:g action on M/N. In the Image φ construct maximal subgroups as normalizers of these diagonals and direct products. Base case: M/N simple, maxes of simple. For the abelian case need complements to abelian chief factors.

54 Find Maximal Subgroups The possible socles arise from chief factors M/N of G. Homomorphism φ:g action on M/N. In the Image φ construct maximal KLEIDMANN, LIEBECK; subgroups as normalizers of these BRAY, HOLT, RONEY-DOUGAL diagonals and direct products. Base case: M/N simple, maxes of simple. For the abelian case need complements to abelian chief factors.

55 Find Maximal Subgroups The possible socles arise from chief factors M/N of G. Homomorphism φ:g action on M/N. In the Image φ construct maximal subgroups as normalizers of these diagonals and direct products. Base case: M/N simple, maxes of simple. For the abelian case need complements to abelian chief factors.

56 All maximal subgroups Remaining (awkward) case: Twisted wreath (Stabilizer in Soc(G) trivial). Difficult complement, only for n>5. Table Lookup/Generator Search for maximals of simple. Similar ideas for maximality test/ intermediate subgroups. GAP-Function: (nonsolvable new, not all) MaximalSubgroupClassReps Hook: Options for O NAN-SCOTT classes.

57 All maximal subgroups Remaining (awkward) case: Twisted wreath (Stabilizer in Soc(G) trivial). Difficult complement, only for n>5. Table Lookup/Generator Search for maximals of simple. Similar ideas for maximality test/ intermediate subgroups. GAP-Function: (nonsolvable new, not all) MaximalSubgroupClassReps Hook: Options for O NAN-SCOTT classes.

58 Downwards Today s standard tool is the (Solvable- G Radical) Trivial-Fitting model: - R=Rad(G) is the largest solvable normal subgroup. (G/Rad is Trivial-Fitting) - S * /Rad=Socle(G/Rad) is direct product of simple nonabelian. - G/S * acts on this socle as outer automorphisms and thus is small (often solvable). S * Rad 1

59 Lifting Subgroups Construct subgroups inductively in G steps over the elementary abelian layers of the radical: Rad M,N G, N/M elementary abelian. Assume subgroups of G/N, determine those of G/M. N M Induction start with G/Rad. 1

60 Lifting Subgroups Construct subgroups inductively in G steps over the elementary abelian layers of the radical: M,N G, N/M elementary abelian. CANNON, Assume subgroups of COX, G/N, HOLT determine those of G/M Induction start with G/Rad. H Rad N M 1

61 Lifting Subgroups Construct subgroups inductively in G steps over the elementary abelian layers of the radical: Rad M,N G, N/M elementary abelian. Assume subgroups of G/N, determine those of G/M. N M Induction start with G/Rad. 1

62 Inductive Step WLOG (description in factor group, work with pre-images and representatives) N G elementary abelian. If U G, then U/U N is complement to N/U N in UN/U N. G NU As N is solvable this can be done N U using cohomology. Further fusion under the normalizer of NU - action on cohomology. U N 1

63 Solvable/ Nonsolvable Case If G is solvable this is all what is needed. GAP-Function: SubgroupsSolvableGroup Hook: Functions to limit cases in which complements are searched for. In the general case one needs initially subgroups of G/Rad. GAP-Function: LatticeViaRadical Hook: Only subgroups within Rad<H G. (No filter function, as hard to specify.)

64 Solvable/ Nonsolvable Case If G is solvable this is all what is needed. GAP-Function: SubgroupsSolvableGroup Hook: Functions to limit cases in which complements are searched for. In the general case one needs initially subgroups of G/Rad. GAP-Function: LatticeViaRadical Hook: Only subgroups within Rad<H G. (No filter function, as hard to specify.)

65 Solvable/ Nonsolvable Case If G is solvable this is all what is needed. GAP-Function: SubgroupsSolvableGroup Hook: Functions to limit cases in which complements are searched for. In the general case one needs initially subgroups of G/Rad. GAP-Function: LatticeViaRadical Hook: Only subgroups within Rad<H G. (No filter function, as hard to specify.)

66 Solvable/ Nonsolvable Case If G is solvable this is all what is needed. GAP-Function: SubgroupsSolvableGroup Hook: Functions to limit cases in which complements are searched for. In the general case one needs initially subgroups of G/Rad. GAP-Function: LatticeViaRadical Hook: Only subgroups within Rad<H G. (No filter function, as hard to specify.)

67 Radical Factor For the initial step, G/Rad, use the structure with large normal G subgroup: SU S * /Rad is direct product of S* U simples. Combine subgroups from simple factors. Any other subgroup U then S U Rad 1 intersects S * in one of its subgroups, again complement.

68 Subgroups of Direct Product S * /Rad=Socle(G/Rad)= Ti is a direct product of simple nonabelian groups. - Any subgroup U S * /Rad projects on each component Ti as a subgroup Vi Ti. - Conjugate Vi belong to conjugate U. - For each factor Ti find list of subgroup classes with cyclic extension/tables. - Possible combinations are Subdirect Products.

69 Subgroups of Direct Product S * /Rad=Socle(G/Rad)= Ti is a direct product of simple nonabelian groups. - Any subgroup U S * /Rad projects on each component Ti as a subgroup Vi Ti. - Conjugate Vi belong to conjugate U. - For each factor Ti find list of subgroup classes with cyclic extension/tables. HOLT - Possible combinations are Subdirect 2010 Products. H. 1996, 2005

70 Subgroups of Direct Product S * /Rad=Socle(G/Rad)= Ti is a direct product of simple nonabelian groups. - Any subgroup U S * /Rad projects on each component Ti as a subgroup Vi Ti. - Conjugate Vi belong to conjugate U. - For each factor Ti find list of subgroup classes with cyclic extension/tables. - Possible combinations are Subdirect Products.

71 Technical DETAILS Subdirect ProductsU Single product. Let U G H with A G, B H chosen and direct factor projections α:u A, β:u B. Then: D=(ker β) α A and E=(ker α) β B. Factor groups A/D and B/E isomorphic. Possible D,E and A/D A 1 D 1 isomorphisms A/D B/E parameterize U. α U β B E ker α ker β 1 1 B/E 1

72 Where to go from here

73 How to use this Try to reduce the search to a subgroup (or automorphism group of the subgroup)? Determine how the subgroups you are searching for behave in relation to normal subgroups: Intersection, Span, Complements.

74 Inclusion Information The only generic lattice structure algorithm I know determines inclusion information by testing inclusion of all conjugates in each subgroup. Can one do better? - Fixed Cosets: U<V g iff VgU=Vg - Maximal subgroups of subgroups - As part of construction?

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