Element structure of groups of order 8: Difference between revisions

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{{group family-specific information|
group family = groups of order 8|
information type = element structure|
connective = of}}
{| class="sortable" border="1"
{| class="sortable" border="1"
! Group !! Second part of GAP ID !! Hall-Senior number !! Nilpotency class !! Element structure page
! Group !! Second part of GAP ID !! Hall-Senior number !! Nilpotency class !! Element structure page
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==1-isomorphism==
==1-isomorphism==


There are no [[1-isomorphism]]s between non-isomorphic groups of order 8. In fact, no two non-isomorphic groups of order 8 are [[order statistics-equivalent finite groups|order statistics-equivalent]].
There are no [[1-isomorphism of group|1-isomorphism]]s between non-isomorphic groups of order 8. In fact, no two non-isomorphic groups of order 8 are [[order statistics-equivalent finite groups|order statistics-equivalent]].
 
==Order statistics==
 
{{element orders facts to check against}}
 
Here are the statistics for a ''particular'' order.
 
{| class="sortable" border="1"
! 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 <math>n^{th}</math> root'' statistics. The number of <math>n^{th}</math> roots equals the number of elements whose order divides <math>n</math>.
 
{| class="sortable" border="1"
! Group !! Second part of GAP ID !! Hall-Senior number !! Number of first roots !! Number of <math>2^{nd}</math> roots !! Number of <math>4^{th}</math> roots !! Number of <math>8^{th}</math> 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
|}

Revision as of 23:36, 3 November 2010

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 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
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