Element structure of groups of order 8

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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

Conjugacy class sizes

Full listing

FACTS TO CHECK AGAINST FOR CONJUGACY CLASS SIZES AND STRUCTURE:
Divisibility facts: size of conjugacy class divides order of group | size of conjugacy class divides index of center | size of conjugacy class equals index of centralizer
Bounding facts: size of conjugacy class is bounded by order of derived subgroup
Counting facts: number of conjugacy classes equals number of irreducible representations | class equation of a group

Because of the small order, it turns out that the nilpotency class completely determines the number of conjugacy classes of each size. Further information: Nilpotency class and order determine conjugacy class size statistics for groups up to prime-fourth order

Group Second part of GAP ID Hall-Senior number Nilpotency class Conjugacy class sizes as list Number of size 1 conjugacy classes (= order of center) Number of size 2 conjugacy classes Total number of conjugacy classes
cyclic group:Z8 1 3 1 1,1,1,1,1,1,1,1 8 0 8
direct product of Z4 and Z2 2 2 1 1,1,1,1,1,1,1,1 8 0 8
dihedral group:D8 3 4 2 1,1,2,2,2 2 3 5
quaternion group 4 5 2 1,1,2,2,2 2 3 5
elementary abelian group:E8 5 1 1 1,1,1,1,1,1,1,1 8 0 8

Grouping by conjugacy class sizes

Number of size 1 conjugacy classes (= order of center) Number of size 2 conjugacy classes Total number of conjugacy classes Number of groups with these conjugacy class sizes Nilpotency class(es) attained by these groups List of groups List of GAP IDs (second part)
8 0 8 3 1 cyclic group:Z8, direct product of Z4 and Z2, elementary abelian group:E8 1,2,5
2 3 5 2 2 dihedral group:D8, quaternion group 3, 4

Action of automorphisms and endomorphisms

The automorphism group acts on the group, permuting conjugacy classes, and the inner automorphism group sends every element to within its conjugacy class. We thus get an action of the outer automorphism group on the set of conjugacy classes.

In the table below, the column "Sizes of orbits of size 2 conjugacy classes" gives the sizes of the orbits under the action on size 2 conjugacy classes. Each orbit on elements is twice the size, and the row underneath gives that data.

Group Second part of GAP ID Hall-Senior number Nilpotency class Automorphism group Outer automorphism group Sizes of orbits of size 1 conjugacy classes
Sizes of orbits of elements in them
Sizes or orbits of size 2 conjugacy classes
Sizes of orbits of elements in them
Total number of conjugacy classes
Total number of elements
Total number of orbits under automorphism group
cyclic group:Z8 1 3 1 Klein four-group Klein four-group 1,1,2,4
1,1,2,4
no orbits
no orbits
8
8
4
direct product of Z4 and Z2 2 2 1 dihedral group:D8 dihedral group:D8 1,1,2,4
1,1,2,4
no orbits
no orbits
8
8
4
dihedral group:D8 3 4 2 dihedral group:D8 cyclic group:Z2 1,1
1,1
1,2
2,4
5
8
4
quaternion group 4 5 2 symmetric group:S4 symmetric group:S3 1,1
1,1
3
6
5
8
3
elementary abelian group:E8 5 1 1 projective special linear group:PSL(3,2) projective special linear group:PSL(3,2) 1,7
1,7
no orbits
no orbits
8
8
2

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 n^{th} root statistics. The number of n^{th} roots equals the number of elements whose order divides n.

Group Second part of GAP ID Hall-Senior number Number of first roots Number of 2^{nd} roots Number of 4^{th} roots Number of 8^{th} 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