# Abstract Algebra with GAP

## GAP log - Friday July 18, 2:30 pm - 4:00 pm

```gap> G:= Subgroup(SymmetricGroup(6), [(1,2,3)(4,5,6), (1,4)(3,5)(2,6)]);
Group([ (1,2,3)(4,5,6), (1,4)(2,6)(3,5) ])
gap> Elements(G);
[ (), (1,2,3)(4,5,6), (1,3,2)(4,6,5), (1,4)(2,6)(3,5), (1,5)(2,4)(3,6), (1,6)(2,5)(3,4) ]
gap> d3:= SymmetricGroup(3);
Sym( [ 1 .. 3 ] )
gap> IsmorphismGroup(d3,G);
Variable: 'IsmorphismGroup' must have a value

gap> IsomorphismGroup(d3,G);
Variable: 'IsomorphismGroup' must have a value

gap> IsomorphismGroups(d3,G);
[ (1,2,3), (1,2) ] -> [ (1,2,3)(4,5,6), (1,5)(2,4)(3,6) ]
gap> G:= Subgroup(SymmetricGroup(8), [(1,2,3,4)(5,6,7,8), (1,8)(4,5)(3,6)(2,7)]);
Group([ (1,2,3,4)(5,6,7,8), (1,8)(2,7)(3,6)(4,5) ])
gap> Size(G);
8
gap> d4:= DihedralGroup(IsPermGroup,8);
Group([ (1,2,3,4), (2,4) ])
gap> IsomorphismGroups(d4,G);
[ (1,2,3,4), (2,4) ] -> [ (1,2,3,4)(5,6,7,8), (1,8)(2,7)(3,6)(4,5) ]
gap> G:= Subgroup(SymmetricGroup(6), [(1,2,3)(4,5,6), (1,4)(3,5)(2,6), (2,3)(5,6)]);
Group([ (1,2,3)(4,5,6), (1,4)(2,6)(3,5), (2,3)(5,6) ])
gap> Size(G);
12
gap> orderFrequency(G);
[ [ 1, 1 ], [ 2, 7 ], [ 3, 2 ], [ 6, 2 ] ]
gap> d6:= DihedralGroup(IsPermGroup,12);
Group([ (1,2,3,4,5,6), (2,6)(3,5) ])
gap> IsomorphismGroups(d6,G);
[ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (1,6,2,4,3,5), (1,6)(2,5)(3,4) ]
gap> IsomorphismGroups(G,d6);
[ (1,2,3)(4,5,6), (1,4)(2,6)(3,5), (2,3)(5,6) ] -> [ (1,5,3)(2,6,4), (2,6)(3,5), (1,4)(2,3)(5,6) ]
gap> G:= Subgroup(SymmetricGroup(10), [(1,2,3)(4,5,6), (1,4)(3,5)(2,6), (
> 2,3)(5,6)]);
Group([ (1,2,3)(4,5,6), (1,4)(2,6)(3,5), (2,3)(5,6) ])
gap> G:= Subgroup(SymmetricGroup(10), [(1,2,3,4,5)(6,7,8,9,10), (5,10)(4,6)(1,9)(3,7)(2,8), (1,5)(2,4)(6,10)((7,9)]);
Syntax error: ) expected
G:= Subgroup(SymmetricGroup(10), [(1,2,3,4,5)(6,7,8,9,10), (5,10)(4,6)(1,9)(3,7)(2,8), (1,5)(2,4)(6,10)((7,9)]);
^
gap> G:= Subgroup(SymmetricGroup(10), [(1,2,3,4,5)(6,7,8,9,10), (5,10)(4,6)(1,9)(3,7)(2,8), (1,5)(2,4)(6,10)((7,9)]));
Syntax error: ) expected
G:= Subgroup(SymmetricGroup(10), [(1,2,3,4,5)(6,7,8,9,10), (5,10)(4,6)(1,9)(3,7)(2,8), (1,5)(2,4)(6,10)((7,9)]));
^
gap> G:= Subgroup(SymmetricGroup(10), [(1,2,3,4,5)(6,7,8,9,10), (5,10)(4,6)(1,9)(3,7)(2,8), (1,5)(2,4)(6,10)((7,9)]));
Syntax error: ) expected
G:= Subgroup(SymmetricGroup(10), [(1,2,3,4,5)(6,7,8,9,10), (5,10)(4,6)(1,9)(3,7)(2,8), (1,5)(2,4)(6,10)((7,9)]));
^
gap> G:= Subgroup(SymmetricGroup(10), [(1,2,3,4,5)(6,7,8,9,10), (5,10)(4,6)(1,9)(3,7)(2,8), (1,5)(2,4)(6,10)((7,9)]);
Syntax error: ) expected
G:= Subgroup(SymmetricGroup(10), [(1,2,3,4,5)(6,7,8,9,10), (5,10)(4,6)(1,9)(3,7)(2,8), (1,5)(2,4)(6,10)((7,9)]);
^
gap> G:= Subgroup(SymmetricGroup(10), [(1,2,3,4,5)(6,7,8,9,10), (5,10)(4,6)(1,9)(3,7)(2,8), (1,5)(2,4)(6,10)(7,9)]);
Group([ (1,2,3,4,5)(6,7,8,9,10), (1,9)(2,8)(3,7)(4,6)(5,10), (1,5)(2,4)(6,10)(7,9) ])
gap> Size(G);
20
gap> G:= Subgroup(SymmetricGroup(8), [(1,2,3,4)(5,6,7,8), (1,8)(4,5)(3,6)(2,7), (1,4)(2,3)(5,8)(6,7)]);
Group([ (1,2,3,4)(5,6,7,8), (1,8)(2,7)(3,6)(4,5), (1,4)(2,3)(5,8)(6,7) ])
gap> Size(G);
16
gap> d8:= DihedralGroup(IsPermGroup,16);
Group([ (1,2,3,4,5,6,7,8), (2,8)(3,7)(4,6) ])
gap> IsomorphismGroups(G,d8);
fail
gap> orderFrequency(G);
[ [ 1, 1 ], [ 2, 11 ], [ 4, 4 ] ]
gap> orderFrequency(d8);
[ [ 1, 1 ], [ 2, 9 ], [ 4, 2 ], [ 8, 4 ] ]
gap> NumberSmallGroups(16);
14
gap> e:= AllGroups(Size, 16, IsAbelian, false);
[ <pc group of size 16 with 4 generators>, <pc group of size 16 with 4 generators>, <pc group of size 16 with 4 generators>,
<pc group of size 16 with 4 generators>, <pc group of size 16 with 4 generators>, <pc group of size 16 with 4 generators>,
<pc group of size 16 with 4 generators>, <pc group of size 16 with 4 generators>, <pc group of size 16 with 4 generators> ]
gap> e[1];
<pc group of size 16 with 4 generators>
gap> Elements(e[1]);
[ <identity> of ..., f1, f2, f3, f4, f1*f2, f1*f3, f1*f4, f2*f3, f2*f4, f3*f4, f1*f2*f3, f1*f2*f4, f1*f3*f4, f2*f3*f4, f1*f2*f3*f4 ]
gap> IsomorphismGroups(G,e[1]);
fail
gap> IsomorphismGroups(G,e[2]);
fail
gap> IsomorphismGroups(G,e[3]);
fail
gap> IsomorphismGroups(G,e[4]);
fail
gap> IsomorphismGroups(G,e[5]);
fail
gap> IsomorphismGroups(G,e[6]);
fail
gap> IsomorphismGroups(G,e[7]);
[ (1,2,3,4)(5,6,7,8), (1,8)(2,7)(3,6)(4,5), (1,4)(2,3)(5,8)(6,7) ] -> [ f1*f2*f3, f1*f3, f1*f4 ]
gap> Elements(e[7]);
[ <identity> of ..., f1, f2, f3, f4, f1*f2, f1*f3, f1*f4, f2*f3, f2*f4, f3*f4, f1*f2*f3, f1*f2*f4, f1*f3*f4, f2*f3*f4, f1*f2*f3*f4 ]
gap> HELP("AsPermGroup");
Help: no matching entry found
gap> IsomorphismPermGroup(e[7]);
<action isomorphism>
gap> Elements(IsomorphismPermGroup(e[7]));
Error, no method found! For debugging hints type ?Recovery from NoMethodFound
Error, no 1st choice method found for `AsSSortedList' on 1 arguments called from
AsSSortedList( coll ) called from
<function>( <arguments> ) called from read-eval-loop
you can 'quit;' to quit to outer loop, or
you can 'return;' to continue
brk> gap> Image(IsomorphismPermGroup(e[7]));
Group([ (1,2)(3,13)(4,7)(5,8)(6,10)(9,16)(11,14)(12,15), (1,3)(2,6)(4,9)(5,10)(7,12)(8,13)(11,15)(14,16),
(1,4)(2,7)(3,9)(5,11)(6,12)(8,14)(10,15)(13,16), (1,5)(2,8)(3,10)(4,11)(6,13)(7,14)(9,15)(12,16) ])
gap> h:=Image(IsomorphismPermGroup(e[7]));
Group([ (1,2)(3,13)(4,7)(5,8)(6,10)(9,16)(11,14)(12,15), (1,3)(2,6)(4,9)(5,10)(7,12)(8,13)(11,15)(14,16),
(1,4)(2,7)(3,9)(5,11)(6,12)(8,14)(10,15)(13,16), (1,5)(2,8)(3,10)(4,11)(6,13)(7,14)(9,15)(12,16) ])
gap> IsomorphismGroups(G,h);
[ (1,2,3,4)(5,6,7,8), (1,8)(2,7)(3,6)(4,5), (1,4)(2,3)(5,8)(6,7) ] -> [ (1,12,5,16)(2,9,8,15)(3,14,10,7)(4,6,11,13),
(1,15)(2,16)(3,11)(4,10)(5,9)(6,14)(7,13)(8,12), (1,3)(2,6)(4,9)(5,10)(7,12)(8,13)(11,15)(14,16) ]
gap> S:= SymmetricGroup(3);
Sym( [ 1 .. 3 ] )
gap> DerivedSeries(S);
[ Group([ (2,3), (1,3,2) ]), Group([ (1,3,2) ]), Group(()) ]
gap> IsSolvable(S);
true
gap> d30:= DihedralGroup(IsPermGroup, 60);;
gap> DerivedSeries(d30);
[ Group([ (2,30)(3,29)(4,28)(5,27)(6,26)(7,25)(8,24)(9,23)(10,22)(11,21)(12,20)(13,19)(14,18)(15,17), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,
16,17,18,19,20,21,22,23,24,25,26,27,28,29,30), (1,3,5,7,9,11,13,15,17,19,21,23,25,27,29)(2,4,6,8,10,12,14,16,18,20,22,24,26,28,30)
]), Group([ (1,3,5,7,9,11,13,15,17,19,21,23,25,27,29)(2,4,6,8,10,12,14,16,18,20,22,24,26,28,30) ]), Group(()) ]
gap> IsNilpotentGroup(d30);
false
gap> HELP("LowerCentralSeriesOfGroup");
Help: Showing `Reference: LowerCentralSeriesOfGroup'
gap> LogTo();
```