Right Bol Loop 16.9.4.45 of order 16


0123456789101112131415
1091110131415122348567
2901011121514131435876
3101109151312144216758
4111090141213153127685
5121315149111008762134
6151413121090117583412
7141512131109106854321
8131214150101195671243
9214387650111013121514
1034217586119014151312
1143126857100915141213
1258761342131415091110
1385672431121514901011
1476853214151312101109
1567584123141213111090

Centre:   0   9

Centrum:   0   9   10   11

Nucleus:   0   9

Left Nucleus:   0   9   10   11

Middle Nucleus:   0   9

Right Nucleus:   0   9


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

9 Elements of order 2:   1   2   3   4   9   12   13   14   15

6 Elements of order 4:   5   6   7   8   10   11

Commutator Subloop:   0   9

Associator Subloop:   0   9

2 Conjugacy Classes of size 1:

7 Conjugacy Classes of size 2:

Automorphic Inverse Property:   FAILS.   (1-1)(4-1) neq (1*4)-1

Al Property:   FAILS. The left inner mapping L1,1 = (3,4)(10,11) is not an automorphism.   L1,1(3*5) neq L1,1(3)*L1,1(5)

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

Right (Left, Full) Mult Group Orders:   64 (1024, 2048)


/ revised October, 2001