Element structure of groups of order 8
This article gives specific information, namely, element structure, about a family of groups, namely: groups of order 8.
View element structure of group families | View element structure of groups of a particular order |View other specific information about groups of order 8
| Group | Second part of GAP ID | Hall-Senior number | Nilpotency class | Element structure page |
|---|---|---|---|---|
| cyclic group:Z8 | 1 | 3 | 1 | element structure of cyclic group:Z8 |
| direct product of Z4 and Z2 | 2 | 2 | 1 | element structure of direct product of Z4 and Z2 |
| dihedral group:D8 | 3 | 4 | 2 | element structure of dihedral group:D8 |
| quaternion group | 4 | 5 | 2 | element structure of quaternion group |
| elementary abelian group:E8 | 5 | 1 | 1 | element structure of elementary abelian group:E8 |
1-isomorphism
There are no 1-isomorphisms between non-isomorphic groups of order 8. In fact, no two non-isomorphic groups of order 8 are order statistics-equivalent.
Order statistics
FACTS TO CHECK AGAINST:
ORDER STATISTICS (cf. order statistics, order statistics-equivalent finite groups): number of nth roots is a multiple of n | Finite abelian groups with the same order statistics are isomorphic | Lazard Lie group has the same order statistics as the additive group of its Lazard Lie ring | Frobenius conjecture on nth roots
1-ISOMORPHISM (cf. 1-isomorphic groups): Lazard Lie group is 1-isomorphic to the additive group of its Lazard Lie ring | order statistics-equivalent not implies 1-isomorphic
Here are the statistics for a particular order.
| Group | Second part of GAP ID | Hall-Senior number | Number of elements of order 1 | Number of elements of order 2 | Number of elements of order 4 | Number of elements of order 8 |
|---|---|---|---|---|---|---|
| cyclic group:Z8 | 1 | 3 | 1 | 1 | 2 | 4 |
| direct product of Z4 and Z2 | 2 | 2 | 1 | 3 | 4 | 0 |
| dihedral group:D8 | 3 | 4 | 1 | 5 | 2 | 0 |
| quaternion group | 4 | 5 | 1 | 1 | 6 | 0 |
| elementary abelian group:E8 | 5 | 1 | 1 | 7 | 0 | 0 |
Here are the number of root statistics. The number of roots equals the number of elements whose order divides .
| Group | Second part of GAP ID | Hall-Senior number | Number of first roots | Number of roots | Number of roots | Number of roots |
|---|---|---|---|---|---|---|
| cyclic group:Z8 | 1 | 3 | 1 | 2 | 4 | 8 |
| direct product of Z4 and Z2 | 2 | 2 | 1 | 4 | 8 | 8 |
| dihedral group:D8 | 3 | 4 | 1 | 6 | 8 | 8 |
| quaternion group | 4 | 5 | 1 | 2 | 8 | 8 |
| elementary abelian group:E8 | 5 | 1 | 1 | 8 | 8 | 8 |
