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9 Posets of subgroups of a finite group
9.1 Posets of subgroups
9.1-1 PosetOfSubgroups
‣ PosetOfSubgroups( G ) | ( operation ) |
The poset of subgroups of G.
gap> PosetOfSubgroups(AlternatingGroup(5));
<finite poset of size 59>
gap> PosetOfSubgroups(CyclicGroup(16));
<finite poset of size 5>
9.1-2 PosetOfpSubgroups
‣ PosetOfpSubgroups( G, p ) | ( operation ) |
‣ BrownPoset( G, p ) | ( operation ) |
The poset \mathcal{S}_p(G) of nontrivial p-subgroups of G, also known as the Brown poset of G.
gap> SpG:=BrownPoset(AlternatingGroup(5),2);
<finite poset of size 20>
gap> Core(SpG);
<finite poset of size 5>
9.1-3 ElementaryAbelianpSubgroups
‣ ElementaryAbelianpSubgroups( G, p[, r] ) | ( function ) |
The list of nontrivial elementary abelian p-subgroups of G. If the optional parameter r is given, then it returns the nontrivial elementary abelian p-subgroups of rank at most r.
gap> ElementaryAbelianpSubgroups(AlternatingGroup(5),2);
[ Group([ (1,3)(2,4) ]), Group([ (1,4)(2,3) ]), Group([ (1,2)(3,4) ]), Group([ (1,2)(3,5) ]), Group([ (1,5)(2,3) ]),
Group([ (1,3)(2,5) ]), Group([ (2,5)(3,4) ]), Group([ (2,3)(4,5) ]), Group([ (2,4)(3,5) ]), Group([ (1,3)(4,5) ]),
Group([ (1,5)(3,4) ]), Group([ (1,4)(3,5) ]), Group([ (1,4)(2,5) ]), Group([ (1,5)(2,4) ]), Group([ (1,2)(4,5) ]),
Group([ (1,3)(2,4), (1,2)(3,4) ]), Group([ (1,5)(2,3), (1,2)(3,5) ]), Group([ (2,4)(3,5), (2,3)(4,5) ]),
Group([ (1,5)(3,4), (1,3)(4,5) ]), Group([ (1,4)(2,5), (1,2)(4,5) ]) ]
gap> ElementaryAbelianpSubgroups(AlternatingGroup(5),2,1);
[ Group([ (1,3)(2,4) ]), Group([ (1,4)(2,3) ]), Group([ (1,2)(3,4) ]), Group([ (1,2)(3,5) ]), Group([ (1,5)(2,3) ]),
Group([ (1,3)(2,5) ]), Group([ (2,5)(3,4) ]), Group([ (2,3)(4,5) ]), Group([ (2,4)(3,5) ]), Group([ (1,3)(4,5) ]),
Group([ (1,5)(3,4) ]), Group([ (1,4)(3,5) ]), Group([ (1,4)(2,5) ]), Group([ (1,5)(2,4) ]), Group([ (1,2)(4,5) ]) ]
9.1-4 PosetOfElementaryAbelianpSubgroups
‣ PosetOfElementaryAbelianpSubgroups( G, p ) | ( operation ) |
‣ QuillenPoset( G, p ) | ( operation ) |
The poset \mathcal{A}_p(G) of nontrivial elementary abelian p-subgroups of G, also known as the Quillen poset of G.
gap> ApG:=QuillenPoset(AlternatingGroup(6),2);
<finite poset of size 75>
9.1-5 RadicalpSubgroups
‣ RadicalpSubgroups( G, p ) | ( function ) |
The radical p-subgroups of G. Recall that a p-subgroup H of G is radical if H=O_p(N_G(H)).
gap> RadicalpSubgroups(AlternatingGroup(5),2);
[ Group([ (1,3)(2,4), (1,2)(3,4) ]), Group([ (1,5)(2,3), (1,2)(3,5) ]), Group([ (2,4)(3,5), (2,3)(4,5) ]),
Group([ (1,5)(3,4), (1,3)(4,5) ]), Group([ (1,4)(2,5), (1,2)(4,5) ]) ]
gap> List(last,Size);
[ 4, 4, 4, 4, 4 ]
9.1-6 PosetOfRadicalpSubgroups
‣ PosetOfRadicalpSubgroups( G, p ) | ( operation ) |
‣ BoucPoset( G, p ) | ( operation ) |
The poset \mathcal{B}_p(G) of radical p-subgroups of a finite group G, also known as the Bouc poset.
gap> BpG:=BoucPoset(AlternatingGroup(6),2);
<finite poset of size 75>
gap> SpG:=BrownPoset(AlternatingGroup(6),2);
<finite poset of size 165>
gap> HomotopyEquivalence(SpG,BpG);
fail
9.1-7 RobinsonPoset
‣ RobinsonPoset( G, p ) | ( operation ) |
The points of the Robinson poset \mathcal{R}_p(G) are the chains (H_1 > H_2 > \ldots > H_k) of p-subgroups such that for every i, H_i \triangleleft H_1. The ordering is given by inclusion.
gap> R:=RobinsonPoset(AlternatingGroup(6),2);
<finite poset of size 615>
gap> IsContractible(R);
false
9.1-8 OrbitSubdivisionBoucPoset
‣ OrbitSubdivisionBoucPoset( G, p ) | ( operation ) |
gap> G:=SmallGroup(72,40);
<pc group of size 72 with 5 generators>
gap> B:=OrbitSubdivisionBoucPoset(G,2);
<finite poset of size 5>
gap> IsContractible(B);
true
9.1-9 OrbitSubdivisionQuillenPoset
‣ OrbitSubdivisionQuillenPoset( G, p ) | ( operation ) |
gap> G:=SmallGroup(72,40);
<pc group of size 72 with 5 generators>
gap> A:=OrbitSubdivisionQuillenPoset(G,2);
<finite poset of size 9>
gap> IsContractible(A);
true
9.1-10 OrbitSubdivisionBrownPoset
‣ OrbitSubdivisionBrownPoset( G, p ) | ( operation ) |
gap> G:=SmallGroup(72,40);
<pc group of size 72 with 5 generators>
gap> S:=OrbitSubdivisionBrownPoset(G,2);
<finite poset of size 23>
gap> IsContractible(S);
true
gap> S2:=OrbitSubdivisionBrownPoset(PSL(2,7),2);
<finite poset of size 19>
gap> IsContractible(S2);
false
9.1-11 EulerCharacteristicQuillenPoset
‣ EulerCharacteristicQuillenPoset( G, p ) | ( operation ) |
Returns the Euler characteristic of \mathcal{A}_p(G) without computing the poset. This may be useful when the poset is too big.
9.2 Poset of subspaces of a vector space
9.2-1 PosetOfSubspaces
‣ PosetOfSubspaces( V ) | ( operation ) |
The lattice of subspaces of the finite vector space V.
gap> V:=GF(2)^4;
( GF(2)^4 )
gap> P:=PosetOfSubspaces(V);
<finite poset of size 67>
gap> MaximumPoset(P);
<vector space of dimension 4 over GF(2)>
gap> MinimumPoset(P);
<vector space of dimension 0 over GF(2)>
9.2-2 PosetOfProperSubspaces
‣ PosetOfProperSubspaces( V ) | ( operation ) |
The poset of nontrivial proper subspaces of the finite vector space V.
gap> V:=GF(2)^4;
( GF(2)^4 )
gap> Q:=PosetOfProperSubspaces(V);
<finite poset of size 65>
gap> PosetHomology(Q);
[ [ 0 ], [ ], [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] ]