^{1}

^{2}

This paper computes the group and character table of Trimethylborane and Cyclohaxane. Results show that the groups are isomorphic to the wreath products
*C _{3}wrC_{2}* and

*C*with orders 81 and 384 and with 17 and 28 conjugacy classes respectively, where

_{2}wrC_{6}*C*denotes a cyclic group of order

_{n}*n*.

Symmetry is very important in group theory. The symmetry of non-rigid molecules is a new field in chemistry. Balasubramanian [_{2}H_{2}). He showed that the molecule is non-rigid in that twisting and inversion operations interconvert all the 16 possible conformations into one another. Considering the permutational subgroup of this molecule, he found out that the permutations operations of the non-rigid molecule can be generated by a group product of much simpler groups, known as wreath product and thus it is a group of order 8. Balasubramanian also used the generating function methods to find the irreducible representation of Boron trimethyl B(CH_{3})_{3} and also the non-rigid Deuterion spin species of B(CD_{3})_{3} and also that of Butane. Hamadanian and Ashrafi [_{3})_{3} and prove that is a group of order 1296 with 28 conjugacy classes. Darafsher et al. [

Darafsher et al. [

The character table which is being needed for the classification of wave functions, determination of selection rules, etc. in which were computed for Trimethylborane and Cyclohaxanethe groups are found to be isomorphic to the wreath products C 3 w r C 2 and C 2 w r C 6 with orders 81 and 384 and with 17 and 28 conjugacy classes respectively. We use the GAP package to get the required results.

The next section shows the results that are obtained and some basic definitions while the following section gives the conclusion.

Let Ω be an arbitrary set; we shall often refer to its elements as points. A bijective (a one-to-one, onto mapping) of Ω onto itself is called a permutation of Ω. The set of all permutations of Ω forms a group under composition of mappings, called the Symmetric group of Ω. We shall denote this group by S y m ( Ω ) (or S Ω ), and write S n to denote the special group S y m ( Ω ) when n is a positive integer and Ω = { 1 , 2 , ⋯ , n } . A permutation group is just a subgroup of Symmetric group. If Γ and Δ are nonempty sets, then we call Γ Δ to denote the set of all functions from Δ to Γ. In the case that C is a group, we turn C Δ into a group by defining product “pointwise”

f g ( γ ) : = f ( γ ) g (γ)

for all f , g ∈ C Δ and γ ∈ Δ where the product in the right is in C.

Let C and D be groups and suppose D acts on the nonempty set Δ. Then the wreath product of C by D is defined with respect to this action is defined to be the semidirect product C Δ ⋊ D = C w r D where D acts on the group C Δ via

f d ( γ ) : = f ( γ d − 1 )

for all f ∈ C Δ , γ ∈ Δ and d ∈ D and multiplication for all ( f 1 , d 1 ) , ( f 2 , d 2 ) ∈ C w r D is given by

( f 1 , d 1 ) ( f 2 , d 2 ) = ( f 1 f 2 d 1 − 1 , d 1 d 2 )

(See [

Consider the trimethylborane compound B(CH_{3})_{3} with the structure.

The speediness of the rotation of the methyl group is considered appropriately high that makes the mean time dynamical symmetry of the molecules makes sense. First considering the symmetry of CH_{3} which is a cyclic group of order 3 namely ℤ 3 denoted by A : = 〈 ( 1 , 2 , 3 ) 〉 (as shown in _{3} whose carbon atom is marked as i , 1 ≤ i ≤ 3 . Therefore the full symmetry of trimethylborane is: G = ( B 1 × B 2 × B 3 ) ⋊ A . Which we can write in terms of wreath product as G : = ℤ 3 w r ℤ 3 . We used GAP package to get the group as follows:

(See [

From the foregoing

S/N | Representatives | Size | Name |
---|---|---|---|

1. | () | 1 | 1a |

2. | (7,8,9) | 3 | 3a |

3. | (7,9,8) | 3 | 3b |

4. | (4,5,6) (7,8,9) | 3 | 3c |

5. | (4,5,6) (7,9,8) | 3 | 3d |

6. | (4,6,5) (7,8,9) | 3 | 3e |

7. | (4,6,5) (7,9,8) | 3 | 3f |

8. | (1,4,7) (2,5,8) (3,6,9) | 9 | 3g |

9. | (1,4,7,2,5,8,3,6,9) | 9 | 3h |

10. | (1,4,7,3,6,9,2,5,8) | 9 | 3i |

11. | (1,7,4) (2,8,5) (3,9,6) | 9 | 3j |

12. | (1,4,7) (2,5,8) (3,6,9) | 9 | 3k |

13. | (1,4,7,2,5,8,3,6,9) | 9 | 9a |

14. | (1,4,7,3,6,9,2,5,8) | 9 | 9b |

15. | (1,7,4) (2,8,5) (3,9,6) | 9 | 3l |

16. | (1,7,5,2,8,6,3,9,4) | 9 | 9c |

17. | (1,7,6,3,9,5,2,8,4) | 9 | 9d |

1a | 3a | 3b | 3c | 3d | 3e | 3f | 3g | 3h | 3i | 3j | 3k | 9a | 9b | 3l | 9c | 9d | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

2P | 1a | 3b | 3a | 3f | 3e | 3d | 3c | 3j | 3i | 3h | 3g | 3l | 9d | 9c | 3k | 9b | 9a |

3P | 1a | 1a | 1a | 1a | 1a | 1a | 1a | 1a | 1a | 1a | 1a | 1a | 3g | 3j | 1a | 3g | 3j |

5P | 1a | 3b | 3a | 3f | 3e | 3d | 3c | 3j | 3i | 3h | 3g | 3l | 9d | 9c | 3k | 9b | 9a |

7P | 1a | 3a | 3b | 3c | 3d | 3e | 3f | 3g | 3h | 3i | 3j | 3k | 9a | 9b | 3l | 9c | 9d |

χ 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

χ 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | A | A | A | A ¯ | A ¯ | A ¯ |

χ 3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | A ¯ | A ¯ | A ¯ | A | A | A |

χ 4 | 1 | A | A ¯ | A ¯ | 1 | 1 | A | 1 | A | A ¯ | 1 | 1 | A | A ¯ | 1 | A | A ¯ |

χ 5 | 1 | A ¯ | A | A | 1 | 1 | A ¯ | 1 | A ¯ | A | 1 | 1 | A ¯ | A | 1 | A ¯ | A |

χ 6 | 1 | A | A ¯ | A ¯ | 1 | 1 | A | 1 | A | A ¯ | 1 | A | A ¯ | 1 | A ¯ | 1 | A |

χ 7 | 1 | A ¯ | A | A | 1 | 1 | A ¯ | 1 | A ¯ | A | 1 | A ¯ | A | 1 | A | 1 | A ¯ |

χ 8 | 1 | A | A ¯ | A ¯ | 1 | 1 | A | 1 | A | A ¯ | 1 | A ¯ | 1 | A | A | A ¯ | 1 |

χ 9 | 1 | A ¯ | A | A | 1 | 1 | A ¯ | 1 | A ¯ | A | 1 | A | 1 | A ¯ | A ¯ | A | 1 |

χ 10 | 3 | B | B ¯ | −C | 0 | 0 | C | D ¯ | B ¯ | −B | D | 0 | 0 | 0 | 0 | 0 | 0 |

χ 11 | 3 | B ¯ | B | C | 0 | 0 | −C | D | −B | B ¯ | D ¯ | 0 | 0 | 0 | 0 | 0 | 0 |

χ 12 | 3 | C | −C | −B | 0 | 0 | − B ¯ | D^{-} | B | B ¯ | D | 0 | 0 | 0 | 0 | 0 | 0 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

χ 13 | 3 | −C | C | − B ¯ | 0 | 0 | -B | D | B ¯ | B | D ¯ | 0 | 0 | 0 | 0 | 0 | 0 |

χ 14 | 3 | −B | − B ¯ | B | 0 | 0 | B ¯ | D | −C | C | D ¯ | 0 | 0 | 0 | 0 | 0 | 0 |

χ 15 | 3 | − B ¯ | −B | B ¯ | 0 | 0 | B | D ¯ | C | −C | D | 0 | 0 | 0 | 0 | 0 | 0 |

χ 16 | 3 | 0 | 0 | 0 | D | D ¯ | 0 | 3 | 0 | 0 | 3 | 0 | 0 | 0 | 0 | 0 | 0 |

χ 17 | 3 | 0 | 0 | 0 | D ¯ | D | 0 | 3 | 0 | 0 | 3 | 0 | 0 | 0 | 0 | 0 | 0 |

where: A = E ( 3 ) 2 = − 1 − S q r t ( − 3 ) 2 ; B = − E ( 3 ) − 2 ∗ E ( 3 ) 2 = 3 + S q r t ( − 3 ) 2 ; C = − E ( 3 ) + E ( 3 ) 2 = − S q r t ( − 3 ) ; D = 3 ∗ E ( 3 ) 2 = ( − 3 − 3 ∗ S q r t ( − 3 ) ) / 2 .

Consider the cyclohexane compound C_{6}H_{12} with the structure.

The speediness of the rotation of the hexane group is considered suitably high that makes the mean time dynamical symmetry of the molecules makes sense. First considering the symmetry of C_{6} which is a cyclic group of order 6 namely ℤ 6 denoted by A : = 〈 ( 1 , 2 , 3 , 4 , 5 , 6 ) 〉 (as shown in _{2} whose carbon atom is marked as i , 1 ≤ i ≤ 6 . Therefore, the full symmetry of cyclohexane is: G = ( B 1 × B 2 × B 3 × B 4 × B 5 × B 6 ) ⋊ A . Which we can write in terms of wreath product as G : = ℤ 2 w r ℤ 6 . We used GAP package to get the group as follows:

(See [

From the foregoing

S/N | Representatives | Size | Name |
---|---|---|---|

1. | () | 1 | 1a |

2. | (11,12) | 6 | 2a |

3. | (9,10) (11,12) | 6 | 2b |

4. | (7,8) (11,12) | 6 | 2c |

5. | (7,8) (9,10) (11,12) | 6 | 2d |

6. | (5,6) (11,12) | 6 | 2e |

7. | (5,6) (9,10) (11,12) | 6 | 2f |

8. | (5,6) (7,8) (11,12) | 6 | 2g |

9. | (5,6) (7,8) (9,10) (11,12) | 6 | 2h |

10. | (3,4) (7,8) (11,12) | 6 | 2i |

11. | (3,4) (7,8) (9,10) (11,12) | 6 | 2j |

12. | (3,4) (5,6) (9,10) (11,12) | 6 | 2k |

13. | (3,4) (5,6) (7,8) (9,10) (11,12) | 6 | 2l |

14. | (1,2) (3,4) (5,6) (7,8) (9,10) (11,12) | 1 | 2m |

15. | (1,3,5,7,9,11) (2,4,6,8,10,12) | 32 | 6f |

16. | (1,3,5,7,9,11,2,4,6,8,10,12) | 32 | 12b |

17. | (1,5,9) (2,6,10) (3,7,11) (4,8,12) | 16 | 3b |

18. | (1,5,9) (2,6,10) (3,7,11,4,8,12) | 32 | 6d |

19. | (1,5,9,2,6,10) (3,7,11,4,8,12) | 32 | 6e |

20. | (1,7) (2,8) (3,9) (4,10) (5,11) (6,12) | 8 | 2n |

21. | (1,7) (2,8) (3,9) (4,10) (5,11,6,12) | 24 | 4a |

22. | (1,7) (2,8) (3,9,4,10) (5,11,6,12) | 24 | 4b |

23. | (1,7,2,8) (3,9,4,10) (5,11,6,12) | 8 | 4c |

24. | (1,9,5) (2,10,6) (3,11,7) (4,12,8) | 16 | 3a |

25. | (1,9,5) (2,10,6) (3,11,8,4,12,7) | 32 | 6b |

26. | (1,9,6,2,10,5) (3,11,8,4,12,7) | 16 | 6c |

27. | (1,11,9,7,5,3) (2,12,10,8,6,4) | 32 | 6a |

28. | (1,11,10,8,6,4,2,12,9,7,5,3) | 32 | 12a |

1a | 2a | 2b | 2c | 2d | 2e | 2f | 2g | 2h | 2i | 2j | 2k | 2l | 2m | 6a | 12a | 3a | 6b | 6c | 2n | 4a | 4b | 4c | 3b | 6d | 6e | 6f | 12b | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

3P | 1a | 1a | 1a | 1a | 1a | 1a | 1a | 1a | 1a | 1a | 1a | 1a | 1a | 1a | 3a | 6c | 3b | 3b | 3b | 1a | 2e | 2k | 2m | 3a | 3a | 3a | 3b | 6e |

3P | 1a | 2a | 2b | 2c | 2d | 2e | 2f | 2g | 2h | 2i | 2j | 2k | 2l | 2m | 2n | 4c | 1a | 2i | 2m | 2n | 4a | 4b | 4c | 1a | 2i | 2m | 2n | 4c |

5P | 1a | 2a | 2b | 2c | 2d | 2e | 2f | 2g | 2h | 2i | 2j | 2k | 2l | 2m | 6f | 12b | 3b | 6d | 6e | 2n | 4a | 4b | 4c | 3a | 6b | 6c | 6a | 12a |

7P | 1a | 2a | 2b | 2c | 2d | 2e | 2f | 2g | 2h | 2i | 2j | 2k | 2l | 2m | 6a | 12a | 3a | 6b | 6c | 2n | 4a | 4b | 4c | 3b | 6d | 6e | 6f | 12b |

11P | 1a | 2a | 2b | 2c | 2d | 2e | 2f | 2g | 2h | 2i | 2j | 2k | 2l | 2m | 6f | 12b | 3b | 6d | 6e | 2n | 4a | 4b | 4c | 3a | 6b | 6c | 6a | 12a |

χ 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

χ 2 | 1 | −1 | 1 | 1 | −1 | 1 | −1 | −1 | 1 | 1 | −1 | 1 | −1 | 1 | 1 | −1 | 1 | −1 | 1 | −1 | 1 | −1 | 1 | 1 | −1 | 1 | −1 | 1 |

χ 3 | 1 | −1 | 1 | 1 | −1 | 1 | −1 | −1 | 1 | −1 | 1 | 1 | −1 | 1 | 1 | −1 | 1 | −1 | 1 | −1 | 1 | 1 | −1 | 1 | −1 | 1 | 1 | −1 |

χ 4 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | −1 | −1 | 1 | 1 | 1 | −1 | −1 | −1 | −1 | 1 | 1 | 1 | −1 | −1 |

χ 5 | 1 | −1 | 1 | 1 | −1 | 1 | −1 | −1 | 1 | −1 | 1 | 1 | −1 | 1 | A | −A | − A ¯ | A ¯ | − A ¯ | −1 | 1 | −1 | 1 | −A | A | −A | A ¯ | − A ¯ |

χ 6 | 1 | −1 | 1 | 1 | −1 | 1 | −1 | −1 | 11 | −1 | 1 | 1 | −1 | 1 | A ¯ | − A ¯ | −A | A | −A | −1 | 1 | −1 | 1 | − A ¯ | A ¯ | − A ¯ | A | −A |

χ 7 | 1 | −1 | 1 | 1 | −1 | 1 | −1 | −1 | 1 | −1 | 1 | 1 | −1 | 1 | − A ¯ | A ¯ | −A | A | −A | 1 | −1 | 1 | −1 | − A ¯ | A ¯ | − A ¯ | −A | A |

χ 8 | 1 | −1 | 1 | 1 | −1 | 1 | −1 | −1 | 1 | −1 | 1 | 1 | −1 | 1 | −A | A | − A ¯ | A ¯ | − A ¯ | 1 | −1 | 1 | −1 | −A | A | −A | − A ¯ | A ¯ |

χ 9 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | A | A | − A ¯ | − A ¯ | − A ¯ | −1 | −1 | −1 | −1 | −A | −A | −A | A ¯ | A ¯ |

χ 10 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | A ¯ | A ¯ | −A | −A | −A | −1 | −1 | −1 | −1 | − A ¯ | − A ¯ | − A ¯ | A | A |

χ 11 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | − A ¯ | − A ¯ | −A | −A | −A | 1 | 1 | 1 | 1 | − A ¯ | − A ¯ | − A ¯ | −A | −A |

χ 12 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | −A | −A | − A ¯ | − A ¯ | − A ¯ | 1 | 1 | 1 | 1 | −A | −A | −A | − A ¯ | − A ¯ |

χ 13 | 2 | 0 | −2 | 2 | 0 | −2 | 0 | 0 | 2 | 0 | −2 | 2 | 0 | −2 | 0 | 0 | 2 | 0 | −2 | 0 | 0 | 0 | 0 | 2 | 0 | −2 | 0 | 0 |

χ 14 | 2 | 0 | −2 | 2 | 0 | −2 | 0 | 0 | 2 | 0 | −2 | 2 | 0 | −2 | 0 | 0 | B | 0 | −B | 0 | 0 | 0 | 0 | B ¯ | 0 | − B ¯ | 0 | 0 |

χ 15 | 2 | 0 | −2 | 2 | 0 | −2 | 0 | 0 | 2 | 0 | −2 | 2 | 0 | −2 | 0 | 0 | B ¯ | 0 | − B ¯ | 0 | 0 | 0 | 0 | B | 0 | −B | 0 | 0 |

χ 16 | 3 | −1 | −1 | −1 | 3 | 3 | −1 | −1 | −1 | 3 | −1 | 3 | −1 | 3 | 0 | 0 | 0 | 0 | 0 | −3 | 1 | 1 | −3 | 0 | 0 | 0 | 0 | 0 |

χ 17 | 3 | −1 | −1 | −1 | 3 | 3 | −1 | −1 | −1 | 3 | −1 | 3 | −1 | 3 | 0 | 0 | 0 | 0 | 0 | 3 | −1 | −1 | 3 | 0 | 0 | 0 | 0 | 0 |

χ 18 | 3 | 1 | −1 | −1 | −3 | 3 | 1 | 1 | −1 | −3 | −1 | 3 | 1 | 3 | 0 | 0 | 0 | 0 | 0 | −3 | −1 | 1 | 3 | 0 | 0 | 0 | 0 | 0 |

χ 19 | 3 | 1 | −1 | −1 | −3 | 3 | 1 | 1 | −1 | −3 | −1 | 3 | 1 | 3 | 0 | 0 | 0 | 0 | 0 | 3 | 1 | −1 | −3 | 0 | 0 | 0 | 0 | 0 |

χ 20 | 6 | −2 | −2 | 2 | 2 | −2 | 2 | 2 | −2 | −6 | 2 | −2 | −2 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

χ 21 | 6 | −2 | 2 | −2 | −2 | −2 | 2 | 2 | 2 | 6 | −2 | −2 | −2 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

χ 22 | 6 | 2 | −2 | 2 | −2 | −2 | −2 | −2 | −2 | 6 | 2 | −2 | 2 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

χ 23 | 6 | 2 | 2 | −2 | 2 | −2 | −2 | −2 | 2 | −6 | −2 | −2 | 2 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

χ 24 | 6 | 0 | 2 | −2 | 0 | −6 | 0 | 0 | −2 | 0 | 2 | 6 | 0 | −6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

χ 25 | 6 | −4 | 2 | 2 | 0 | 2 | 0 | 0 | −2 | 0 | −2 | −2 | 4 | −6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

χ 26 | 6 | 0 | −2 | −2 | 0 | 2 | −4 | 4 | 2 | 0 | 2 | −2 | 0 | −6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

χ 27 | 6 | 0 | −2 | −2 | 0 | 2 | 4 | −4 | 2 | 0 | 2 | −2 | 0 | −6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

χ 28 | 6 | 4 | 2 | 2 | 0 | 2 | 0 | 0 | −2 | 0 | −2 | −2 | −4 | −6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

where: A = − E ( 3 ) = 1 − S q r t ( − 3 ) 2 ; B = 2 ∗ E ( 3 ) = − 1 + S q r t ( − 3 ) .

In this paper, we computed the group, the conjugacy classes and character tables of Trimethylborane and Cyclohaxane as seen in Tables 1-4. We found that the groups are isomorphic to the wreath products C 3 w r C 2 and C 2 w r C 6 with orders 81 and 384 and with 17 and 28 conjugacy classes respectively, where C n is cyclic group of order n. We used the GAP package for our calculations. The character tables obtained give room for the needed classification of wavefunctions, determination of selection rules, and so on.

The authors declare no conflicts of interest regarding the publication of this paper.

Suleiman, E. and Audu, M.S. (2020) Computing the Full Non-Rigid Group of Trimethylborane and Cyclohaxane Using Wreath Product. American Journal of Computational Mathematics, 10, 23-30. https://doi.org/10.4236/ajcm.2020.101002