Dear GAP Forum,
I try to extend the number of points acted on by permutation groups to
multiples of the original domain.
For example i tried to find a subgroup of S60 isomorphic to A5, thus acting on
60 points.
As the samples attached show, I did it by brute force, extending the generators
by acting on all residue classes mod the multiplier.
This method yields subgroups of the desired target symmetric groups, but only
in one subgroup conjugacy class. The conjugacy class(es) I am looking for, are
those, whose subgroups are transitive, as needed, for example, to be true
rotation groups acting on the vertices of a solid.
This was possible in the case of A5 extending to S12, but not to S30 or even
S60, as those targets are to big for executing, for example IsomorphicSubgroups.
In the case of finite rotation groups there are other methods using
presentations in Linear Algebra too. I used blueprints of solids to get the
right generator permutations for S12 and S60 for the desired rotation groups.
1. Is there a simpler method for extending the domain acted on to a multiple in
general?
2. Is there a feasable method for big Symmetric Groups as S60 to find conjugacy
classes of isomorphic subgroups (for example for A5) beeing transitive?
Thank you and best regards, Rudolf Zlabinger
# Two examples of extending the points acted on to a set of a multiple Size
# The method is to assign the original points the mods 1 of the multiplier
# and extending the generators by cycles consisting of the rest of the mods
# The result yields a isomorphic subgroup of the symmetric group (nr of points)
# It may be not the desired conjugacy class, thus a finite rotation group may
not be
# preserved. In the case of finite rotation groups one has to look for a
# conjugacy class of transitive groups, i.e a group whose orbit on a point is
the full set
# of points
# First we extend a group acting on 12 points to a group acting on 60 points
# Thats the group covering a "canonical" label set of icosahedron
# gencgroup:=GeneratorsOfGroup(cgroup);
# [ (1,2,3)(4,6,7)(5,11,8)(9,10,12), (1,2)(3,6)(4,11)(5,7)(8,10)(9,12) ]
# cgroup itself is not transitive, and therefore itself not a true rotation
group
# we calculate the generators as acting on numbers mod(5), where the mods 1
are the
# original
# points of the icosahedron
# list0010:=[1,2,3,4,6,7,5,11,8,9,10,12];
# list0020:=[1,2,3,6,4,11,5,7,8,10,9,12];
# list0030:=[1,2,3,4,5];
# result0010:=List( list0030,c->List( list0010,d->( (d-1)*5+c) ) );
# result0020:=List( list0030,c->List( list0020,d->( (d-1)*5+c) ) );
# result0010;
# [ [ 1, 6, 11, 16, 26, 31, 21, 51, 36, 41, 46, 56 ],
# [ 2, 7, 12, 17, 27, 32, 22, 52, 37, 42, 47, 57 ],
# [ 3, 8, 13, 18, 28, 33, 23, 53, 38, 43, 48, 58 ],
# [ 4, 9, 14, 19, 29, 34, 24, 54, 39, 44, 49, 59 ],
# [ 5, 10, 15, 20, 30, 35, 25, 55, 40, 45, 50, 60 ] ]
# result0020;
# [ [ 1, 6, 11, 26, 16, 51, 21, 31, 36, 46, 41, 56 ],
# [ 2, 7, 12, 27, 17, 52, 22, 32, 37, 47, 42, 57 ],
# [ 3, 8, 13, 28, 18, 53, 23, 33, 38, 48, 43, 58 ],
# [ 4, 9, 14, 29, 19, 54, 24, 34, 39, 49, 44, 59 ],
# [ 5, 10, 15, 30, 20, 55, 25, 35, 40, 50, 45, 60 ] ]
# So we get as generators of our icosahedron acting on 60 points
gen_0010:= (1, 6, 11) (16, 26, 31) (21, 51, 36) (41, 46, 56)
(2, 7, 12) (17, 27, 32) (22, 52, 37) (42, 47, 57)
(3, 8, 13) (18, 28, 33) (23, 53, 38) (43, 48, 58)
(4, 9, 14) (19, 29, 34) (24, 54, 39) (44, 49, 59)
(5, 10, 15) (20, 30, 35) (25, 55, 40) (45, 50, 60);
gen_0020:= (1, 6) (11, 26) (16, 51) (21, 31) (36, 46) (41, 56)
(2, 7) (12, 27) (17, 52) (22, 32) (37, 47) (42, 57)
(3, 8) (13, 28) (18, 53) (23, 33) (38, 48) (43, 58)
(4, 9) (14, 29) (19, 54) (24, 34) (39, 49) (44, 59)
(5, 10) (15, 30) (20, 55) (25, 35) (40, 50) (45, 60);
group0010:=Group(gen_0010,gen_0020);
# StructureDescription(group0010);
# "A5" It is isomorphic indeed
# NrMovedPoints(group0010);
# 60 and acting on 60 points as desired
# The next sample extends A5 to a isomorphic subgroup of S30
# Generators of A5 (1,3,2), (2,4,3), (2,3)(4,5)
gens0050:= ( 1,13, 7)( 2,14, 8)( 3,15, 9)( 4,16,10)( 5,17,11)( 6,18,12);
gens0051:= ( 7,19,13)( 8,20,14)( 9,21,15)(10,22,16)(11,23,17)(12,24,18);
gens0052:= ( 7,13)( 8,14)( 9,15)(10,16)(11,17)(12,18)
(19,25)(20,26)(21,27)(22,28)(23,29)(24,30);
group0050:=Group(gens0050,gens0051,gens0052);
_______________________________________________
Forum mailing list
Forum@mail.gap-system.org
http://mail.gap-system.org/mailman/listinfo/forum
