[seqfan] Re: Floretions
Richard Mathar
mathar at strw.leidenuniv.nl
Mon May 11 17:43:20 CEST 2009
On Apr 29 Creighton schrieb/schrab/schrub:
> creighton.k.dement at uni-oldenburg.de creighton.k.dement at uni-oldenburg.de
> Wed Apr 29 21:17:28 CEST 2009
>
> Dear Seqfans,
>
> I have uploaded a draft version of my paper on floretion-generated integer sequences to my website (scroll down to bottom of page)
> http://www.fumba.eu/sitelayout/Floretion.html
>
> As this is an entirely private work unaffiliated with a university, I would be
> very happy to receive as many constructive suggestions as possible (depending on
> how well it is received, I may eventually try to get some of it published).
> Currently, there is a fair number of additional conjectures and I am compiling a
> formal list where outside help is definitely needed. That said, several of these
> conjectures are stated in the paper itself.
The list of the members of the group of order 32 is
mem := [ EE, IE, JE, KE, -EE, -IE, -JE, -KE, EI, EJ,
EK, -EI, -EJ, -EK, II, JJ, KK, IJ, IK, JI, JK,
KI, KJ, -II, -JJ, -KK, -IJ, -IK, -JI, -JK, -KI, -KJ]
This yields a 32x32 group multiplication table (to be checked!)
EE IE JE KE -EE -IE -JE -KE EI EJ EK -EI -EJ -EK II JJ KK IJ IK JI JK KI KJ -II -JJ -KK -IJ -IK -JI -JK -KI -KJ
IE -EE KE -JE -IE EE -KE JE II IJ IK -II -IJ -IK -EI KJ -JK -EJ -EK KI KK -JI -JJ EI -KJ JK EJ EK -KI -KK JI JJ
JE -KE -EE IE -JE KE EE -IE JI JJ JK -JI -JJ -JK -KI -EJ IK -KJ -KK -EI -EK II IJ KI EJ -IK KJ KK EI EK -II -IJ
KE JE -IE -EE -KE -JE IE EE KI KJ KK -KI -KJ -KK JI -IJ -EK JJ JK -II -IK -EI -EJ -JI IJ EK -JJ -JK II IK EI EJ
-EE -IE -JE -KE EE IE JE KE -EI -EJ -EK EI EJ EK -II -JJ -KK -IJ -IK -JI -JK -KI -KJ II JJ KK IJ IK JI JK KI KJ
-IE EE -KE JE IE -EE KE -JE -II -IJ -IK II IJ IK EI -KJ JK EJ EK -KI -KK JI JJ -EI KJ -JK -EJ -EK KI KK -JI -JJ
-JE KE EE -IE JE -KE -EE IE -JI -JJ -JK JI JJ JK KI EJ -IK KJ KK EI EK -II -IJ -KI -EJ IK -KJ -KK -EI -EK II IJ
-KE -JE IE EE KE JE -IE -EE -KI -KJ -KK KI KJ KK -JI IJ EK -JJ -JK II IK EI EJ JI -IJ -EK JJ JK -II -IK -EI -EJ
EI II JI KI -EI -II -JI -KI -EE EK -EJ EE -EK EJ -IE JK -KJ IK -IJ -JE -JJ -KE KK IE -JK KJ -IK IJ JE JJ KE -KK
EJ IJ JJ KJ -EJ -IJ -JJ -KJ -EK -EE EI EK EE -EI -IK -JE KI -IE II -JK JI -KK -KE IK JE -KI IE -II JK -JI KK KE
EK IK JK KK -EK -IK -JK -KK EJ -EI -EE -EJ EI EE IJ -JI -KE -II -IE JJ -JE KJ -KI -IJ JI KE II IE -JJ JE -KJ KI
-EI -II -JI -KI EI II JI KI EE -EK EJ -EE EK -EJ IE -JK KJ -IK IJ JE JJ KE -KK -IE JK -KJ IK -IJ -JE -JJ -KE KK
-EJ -IJ -JJ -KJ EJ IJ JJ KJ EK EE -EI -EK -EE EI IK JE -KI IE -II JK -JI KK KE -IK -JE KI -IE II -JK JI -KK -KE
-EK -IK -JK -KK EK IK JK KK -EJ EI EE EJ -EI -EE -IJ JI KE II IE -JJ JE -KJ KI IJ -JI -KE -II -IE JJ -JE KJ -KI
II -EI KI -JI -II EI -KI JI -IE IK -IJ IE -IK IJ EE KK JJ -EK EJ -KE -KJ JE -JK -EE -KK -JJ EK -EJ KE KJ -JE JK
JJ -KJ -EJ IJ -JJ KJ EJ -IJ -JK -JE JI JK JE -JI KK EE II KE -KI EK -EI -IK -IE -KK -EE -II -KE KI -EK EI IK IE
KK JK -IK -EK -KK -JK IK EK KJ -KI -KE -KJ KI KE JJ II EE -JI -JE -IJ IE -EJ EI -JJ -II -EE JI JE IJ -IE EJ -EI
IJ -EJ KJ -JJ -IJ EJ -KJ JJ -IK -IE II IK IE -II EK -KE -JI EE -EI -KK KI JK JE -EK KE JI -EE EI KK -KI -JK -JE
IK -EK KK -JK -IK EK -KK JK IJ -II -IE -IJ II IE -EJ -KI JE EI EE KJ -KE -JJ JI EJ KI -JE -EI -EE -KJ KE JJ -JI
JI -KI -EI II -JI KI EI -II -JE JK -JJ JE -JK JJ KE -EK -IJ -KK KJ EE EJ -IE IK -KE EK IJ KK -KJ -EE -EJ IE -IK
JK -KK -EK IK -JK KK EK -IK JJ -JI -JE -JJ JI JE -KJ EI -IE KI KE -EJ EE IJ -II KJ -EI IE -KI -KE EJ -EE -IJ II
KI JI -II -EI -KI -JI II EI -KE KK -KJ KE -KK KJ -JE -IK EJ JK -JJ IE IJ EE -EK JE IK -EJ -JK JJ -IE -IJ -EE EK
KJ JJ -IJ -EJ -KJ -JJ IJ EJ -KK -KE KI KK KE -KI -JK IE -EI -JE JI IK -II EK EE JK -IE EI JE -JI -IK II -EK -EE
-II EI -KI JI II -EI KI -JI IE -IK IJ -IE IK -IJ -EE -KK -JJ EK -EJ KE KJ -JE JK EE KK JJ -EK EJ -KE -KJ JE -JK
-JJ KJ EJ -IJ JJ -KJ -EJ IJ JK JE -JI -JK -JE JI -KK -EE -II -KE KI -EK EI IK IE KK EE II KE -KI EK -EI -IK -IE
-KK -JK IK EK KK JK -IK -EK -KJ KI KE KJ -KI -KE -JJ -II -EE JI JE IJ -IE EJ -EI JJ II EE -JI -JE -IJ IE -EJ EI
-IJ EJ -KJ JJ IJ -EJ KJ -JJ IK IE -II -IK -IE II -EK KE JI -EE EI KK -KI -JK -JE EK -KE -JI EE -EI -KK KI JK JE
-IK EK -KK JK IK -EK KK -JK -IJ II IE IJ -II -IE EJ KI -JE -EI -EE -KJ KE JJ -JI -EJ -KI JE EI EE KJ -KE -JJ JI
-JI KI EI -II JI -KI -EI II JE -JK JJ -JE JK -JJ -KE EK IJ KK -KJ -EE -EJ IE -IK KE -EK -IJ -KK KJ EE EJ -IE IK
-JK KK EK -IK JK -KK -EK IK -JJ JI JE JJ -JI -JE KJ -EI IE -KI -KE EJ -EE -IJ II -KJ EI -IE KI KE -EJ EE IJ -II
-KI -JI II EI KI JI -II -EI KE -KK KJ -KE KK -KJ JE IK -EJ -JK JJ -IE -IJ -EE EK -JE -IK EJ JK -JJ IE IJ EE -EK
-KJ -JJ IJ EJ KJ JJ -IJ -EJ KK KE -KI -KK -KE KI JK -IE EI JE -JI -IK II -EK -EE -JK IE -EI -JE JI IK -II EK EE
Supposed this multiplication table is correct,
the orders of the elements (the powers that evaluate to the unit element EE) are
IE^4, JE^4, KE^4, -EE^2, -IE^4, -JE^4, -KE^4, EI^4, EJ^4, EK^4,
-EI^4, -EJ^4, -EK^4, II^2, JJ^2, KK^2, IJ^2, IK^2, JI^2, JK^2,
KI^2, KJ^2, -II^2, -JJ^2, -KK^2, -IJ^2, -IK^2, -JI^2, -JK^2,
-KI^2, -KJ^2,
If we feed the full multiplication table (converted to 31 generators)
into GAP4 ( http://www.gap-system.org/ ) it says
gap> Order(g) ;
32
gap> StructureDescription(g) ;
"(C2 x D8) : C2"
So the structure of the group is the semi-direct product of a group
build by (a direct product of the cyclic group C2 by the dihedral group D8)
multiplied by the cyclic group C2.
gap> FittingSubgroup(g) ;
Group([ f1, f2, f8, f9 ])
The largest nilpotent normal subgroup is given by the generators IE, JE, EI, EJ.
To display the character table:
gap> c := CharacterTable(g) ;
CharacterTable( <fp group of size 32 with 31 generators> )
gap> Display(c) ;
CT1
2 5 4 4 4 4 5 4 4 4 4 4 4 4 4 4 4 4
1a 4a 4b 4c 4d 2a 4e 2b 2c 2d 2e 4f 2f 2g 2h 2i 2j
X.1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
X.2 1 -1 -1 1 -1 1 1 -1 1 -1 1 -1 1 -1 1 1 -1
X.3 1 1 1 -1 1 1 1 -1 1 -1 1 -1 -1 1 -1 -1 -1
X.4 1 -1 -1 -1 -1 1 1 1 1 1 1 1 -1 -1 -1 -1 1
X.5 1 1 -1 -1 -1 1 -1 -1 -1 1 1 1 1 1 1 -1 -1
X.6 1 -1 1 -1 1 1 -1 1 -1 -1 1 -1 1 -1 1 -1 1
X.7 1 1 -1 1 -1 1 -1 1 -1 -1 1 -1 -1 1 -1 1 1
X.8 1 -1 1 1 1 1 -1 -1 -1 1 1 1 -1 -1 -1 1 -1
X.9 1 -1 -1 1 1 1 1 -1 -1 -1 -1 1 1 1 -1 -1 1
X.10 1 1 1 1 -1 1 1 1 -1 1 -1 -1 1 -1 -1 -1 -1
X.11 1 -1 -1 -1 1 1 1 1 -1 1 -1 -1 -1 1 1 1 -1
X.12 1 1 1 -1 -1 1 1 -1 -1 -1 -1 1 -1 -1 1 1 1
X.13 1 -1 1 -1 -1 1 -1 1 1 -1 -1 1 1 1 -1 1 -1
X.14 1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 -1 -1 1 1
X.15 1 -1 1 1 -1 1 -1 -1 1 1 -1 -1 -1 1 1 -1 1
X.16 1 1 -1 1 1 1 -1 1 1 -1 -1 1 -1 -1 1 -1 -1
X.17 4 . . . . -4 . . . . . . . . . . .
This is also listed as number 49 in the small groups table of
H. U. Besche et al ( http://www-public.tu-bs.de:8080/~hubesche/small.html )
gap> s := SmallGroup(32,49) ;
gap> StructureDescription(s) ;
"(C2 x D8) : C2"
Since this is basically an attempt to build an algebra with this group,
this information may help to track down more information in the literature:
see
David J. Rusin, The Cohomology of the groups of Order 32, Math. Comp. 53 (no 187) (1989) 359-385
http://www.jstor.org/stable/2008369
The group seems to be listed as T22 in Table 8A of
G. Butler, J. McKay, The transitive groups of degree up to eleven,
Commun. in Algebra, Vol 11 (8) (1983) 863-911, DOI: 10.1080/00927878308822884
from one we may get the generators in permutation form.
Helpful also
Marcel Wild, The groups of order sixteen made easy, Am. Math. Monthly 112 (1) (2005) 20-31
http://www.jstor.org/stable/30037381
More information about the SeqFan
mailing list