Right Bol Loop 16.5.2.4 of order 16


0123456789101112131415
1101190141512132348765
2111009151413121437856
3901110121315144216578
4091011131214153125687
5151413121110096784321
6141512131011905873412
7131214150910118652134
8121315149011107561243
9214365870111013121514
1034217865119014151312
1143128756100915141213
1256781234131514091110
1365872143121415901011
1487564312151213101109
1578653421141312111090

Centre:   0   9

Centrum:   0   9

Nucleus:   0   9

Left Nucleus:   0   9   12   13

Middle Nucleus:   0   9   10   11

Right Nucleus:   0   9   10   11


Comm(L):   This graph has as its 7 vertices the nontrivial cosets of the centre. Edges represent non-commuting cosets. Here we print (in reverse video) the complementary graph, in which edges represent commuting cosets.


1 Element of order 1:   0

5 Elements of order 2:   9   12   13   14   15

2 Elements of order 4:   10   11

8 Elements of order 8:   1   2   3   4   5   6   7   8

Commutator Subloop:   0   9   10   11

Associator Subloop:   0   9   10   11

2 Conjugacy Classes of size 1:

1 Conjugacy Class of size 2:

3 Conjugacy Classes of size 4:

Automorphic Inverse Property:   FAILS.   (1-1)(8-1) neq (1*8)-1

Al Property:   FAILS. The left inner mapping L1,1 = (5,7,6,8)(12,14,13,15) is not an automorphism.   L1,1(1*5) neq L1,1(1)*L1,1(5)

Ar Property:   HOLDS (i.e. every right inner mapping Ra,b is an automorphism)

Right (Left, Full) Mult Group Orders:   32 (512, 2048)


/ revised October, 2001