Element structure of groups of order 81: Difference between revisions

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| [[Direct product of Z27 and Z3]] || 5 || 1 || [[element structure of direct product of Z27 and Z3]]
| [[Direct product of Z27 and Z3]] || 5 || 1 || [[element structure of direct product of Z27 and Z3]]
|-
|-
| [[Semidirect product of Z27 and Z3]] || 6 || 2 || [[element structure of semidirect product of Z27 and Z3]]
| [[M81]] (semidirect product of Z27 and Z3)|| 6 || 2 || [[element structure of M81]]
|-
|-
| [[Wreath product of Z3 and Z3]] || 7 || 3 || [[element structure of wreath product of Z3 and Z3]]
| [[Wreath product of Z3 and Z3]] || 7 || 3 || [[element structure of wreath product of Z3 and Z3]]
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===Pairs where one of the groups is abelian===
===Pairs where one of the groups is abelian===


Of the 15 [[groups of order 81]], 5 are abelian, 6 have nilpotency class two, and 4 have nilpotency class three. Via the [[Baer correspondence]], each of the groups of class two has a Baer Lie ring, and in particular is 1-isomorphic to the additive group of that Lie ring. Of the 4 groups of nilpotency class three, only one ([[SmallGroup(81,8)]]) is 1-isoomorphic to an abelian group. There are ''no 1-isomorphisms between pairs where both members are non-abelian.''
Of the 15 [[groups of order 81]], 5 are abelian, 6 have nilpotency class two, and 4 have nilpotency class three. Via the [[Baer correspondence]], each of the groups of class two has a Baer Lie ring, and in particular is 1-isomorphic to the additive group of that Lie ring. Of the 4 groups of nilpotency class three, only one ([[SmallGroup(81,8)]]) is 1-isomorphic to an abelian group. There are ''no 1-isomorphisms between pairs where both members are non-abelian.''


{| class="sortable" border="1"
{| class="sortable" border="1"
! Non-abelian member of pair !! Nilpotency class !! GAP ID !! Abelian member of pair !! GAP ID !! Nature of the 1-isomorphism !! Description of the 1-isomorphism !! Best perspective !! Alternative perspective
! Non-abelian member of pair !! Nilpotency class !! GAP ID !! Abelian member of pair !! GAP ID !! Nature of the 1-isomorphism !! Description of the 1-isomorphism !! Best perspective !! Alternative perspective
|-
|-
| [[semidirect product of Z27 and Z3]] || 2 || 6 || [[direct product of Z27 and Z3]] || 5 || [[Baer correspondence between odd-order class two p-groups and odd-order class two p-Lie rings]] || [[Baer correspondence between semidirect product of Z27 and Z3 and its Lie ring]] || ||
| [[M81]] || 2 || 6 || [[direct product of Z27 and Z3]] || 5 || [[Baer correspondence]] || [[Baer correspondence between M81 and its Lie ring]] || ||
|-
|-
| [[nontrivial semidirect product of Z9 and Z9]] || 2 || 4 || [[direct product of Z9 and Z9]] || 2 || [[Baer correspondence between odd-order class two p-groups and odd-order class two p-Lie rings]] || [[Baer correspondence between nontriival semidirect product of Z9 and Z9 and its Lie ring]] || ||
| [[nontrivial semidirect product of Z9 and Z9]] || 2 || 4 || [[direct product of Z9 and Z9]] || 2 || [[Baer correspondence ]] || [[Baer correspondence between nontriival semidirect product of Z9 and Z9 and its Lie ring]] || ||
|-
|-
| [[SmallGroup(81,3)]] || 2 || 3 || [[direct product of Z9 and E9]] || 11 || [[Baer correspondence between odd-order class two p-groups and odd-order class two p-Lie rings]] || [[Baer correspondence between SmallGroup(81,3) and its Lie ring]] || ||
| [[SmallGroup(81,3)]] || 2 || 3 || [[direct product of Z9 and E9]] || 11 || [[Baer correspondence]] || [[Baer correspondence between SmallGroup(81,3) and its Lie ring]] || ||
|-
|-
| [[direct product of semidirect product of Z9 and Z3 and Z3]] || 2 || 13 || [[direct product of Z9 and E9]] || 11 || [[Baer correspondence between odd-order class two p-groups and odd-order class two p-Lie rings]] || [[Baer correspondence between direct product of semidirect product of Z9 and Z3 and Z3 and its Lie ring]] || ||
| [[direct product of semidirect product of Z9 and Z3 and Z3]] || 2 || 13 || [[direct product of Z9 and E9]] || 11 || [[Baer correspondence]] || [[Baer correspondence between direct product of semidirect product of Z9 and Z3 and Z3 and its Lie ring]] || ||
|-
|-
| [[central product of prime-cube order group:U(3,3) and Z9]] || 2 || 14 || [[direct product of Z9 and E9]] || 11 || [[Baer correspondence between odd-order class two p-groups and odd-order class two p-Lie rings]] || [[Baer correspondence between central product of prime-cube order group:U(3,3) and Z9 and its Lie ring]] || ||
| [[central product of prime-cube order group:U(3,3) and Z9]] || 2 || 14 || [[direct product of Z9 and E9]] || 11 || [[Baer correspondence]] || [[Baer correspondence between central product of prime-cube order group:U(3,3) and Z9 and its Lie ring]] || ||
|-
|-
| [[direct product of prime-cube order group:U(3,3) and Z3]] || 2 || 12 || [[elementary abelian group:E81]] || 15 || [[Baer correspondence between odd-order class two p-groups and odd-order class two p-Lie rings]] || [[Baer correspondence between direct product of prime-cube order group:U(3,3) and Z3 and its Lie ring]] || ||
| [[direct product of prime-cube order group:U(3,3) and Z3]] || 2 || 12 || [[elementary abelian group:E81]] || 15 || [[Baer correspondence]] || [[Baer correspondence between direct product of prime-cube order group:U(3,3) and Z3 and its Lie ring]] || ||
|-
|-
| [[SmallGroup(81,8)]] || 3 || 8 || [[direct product of Z9 and E9]] || 11 || ? || || ||
| [[SmallGroup(81,8)]] || 3 || 8 || [[direct product of Z9 and E9]] || 11 || ? || || ||
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| [[direct product of Z9 and Z9]] || 2 || 2 || [[nontrivial semidirect product of Z9 and Z9]] || 4
| [[direct product of Z9 and Z9]] || 2 || 2 || [[nontrivial semidirect product of Z9 and Z9]] || 4
|-
|-
| [[direct product of Z27 and Z3]] || 5 || 2 || [[semidirect product of Z27 and Z3]] || 6
| [[direct product of Z27 and Z3]] || 5 || 2 || [[M81]] || 6
|-
|-
| [[direct product of Z9 and E9]] || 11 || 5 || [[SmallGroup(81,3)]], [[SmallGroup(81,8)]], [[direct product of semidirect product of Z9 and Z3 and Z3]], [[central product of prime-cube order group:U(3,3) and Z9]] || 3, 8, 13, 14
| [[direct product of Z9 and E9]] || 11 || 5 || [[SmallGroup(81,3)]], [[SmallGroup(81,8)]], [[direct product of semidirect product of Z9 and Z3 and Z3]], [[central product of prime-cube order group:U(3,3) and Z9]] || 3, 8, 13, 14
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| [[Direct product of Z27 and Z3]] || 5 || 1 || 8 || 18 || 54 || 0
| [[Direct product of Z27 and Z3]] || 5 || 1 || 8 || 18 || 54 || 0
|-
|-
| [[Semidirect product of Z27 and Z3]] || 6 || 1 || 8 || 18 || 54 || 0
| [[M81]] || 6 || 1 || 8 || 18 || 54 || 0
|-
|-
| [[Wreath product of Z3 and Z3]] || 7 || 1 || 44 || 36 || 0 || 0
| [[Wreath product of Z3 and Z3]] || 7 || 1 || 44 || 36 || 0 || 0
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| [[Direct product of Z27 and Z3]] || 5 || 1 || 9 || 27 || 81 || 81
| [[Direct product of Z27 and Z3]] || 5 || 1 || 9 || 27 || 81 || 81
|-
|-
| [[Semidirect product of Z27 and Z3]] || 6 || 1 || 9 || 27 || 81 || 81
| [[M81]] || 6 || 1 || 9 || 27 || 81 || 81
|-
|-
| [[Wreath product of Z3 and Z3]] || 7 || 1 || 45 || 81 || 81 || 81
| [[Wreath product of Z3 and Z3]] || 7 || 1 || 45 || 81 || 81 || 81
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| 1,2,6,18,54 || 1,3,9,27,81 || 1 || 1 || [[Cyclic group:Z81]] (ID:1) || --  || Yes || Yes
| 1,2,6,18,54 || 1,3,9,27,81 || 1 || 1 || [[Cyclic group:Z81]] (ID:1) || --  || Yes || Yes
|-
|-
| 1,8,18,54,0 || 1,9,27,81,81 || 2 || 1 || [[Direct product of Z27 and Z3]] (ID:5) and [[Semidirect product of Z27 and Z3]] (ID:6) || -- || Yes || Yes
| 1,8,18,54,0 || 1,9,27,81,81 || 2 || 1 || [[Direct product of Z27 and Z3]] (ID:5) and [[M81]] (ID:6) || -- || Yes || Yes
|-
|-
| 1,8,72,0,0 || 1,9,81,81,81 || 3 || 2 || [[Direct product of Z9 and Z9]] (ID:2), [[Semidirect product of Z9 and Z9]] (ID:4) || [[SmallGroup(81,10)]] (ID:10) || Yes || Yes
| 1,8,72,0,0 || 1,9,81,81,81 || 3 || 2 || [[Direct product of Z9 and Z9]] (ID:2), [[Semidirect product of Z9 and Z9]] (ID:4) || [[SmallGroup(81,10)]] (ID:10) || Yes || Yes

Latest revision as of 02:09, 20 July 2011

Group Second part of GAP ID Nilpotency class Element structure page
Cyclic group:Z81 1 1 element structure of cyclic group:Z81
Direct product of Z9 and Z9 2 1 element structure of direct product of Z9 and Z9
SmallGroup(81,3) 3 2 element structure of SmallGroup(81,3)
Nontrivial semidirect product of Z9 and Z9 4 2 element structure of nontrivial semidirect product of Z9 and Z9
Direct product of Z27 and Z3 5 1 element structure of direct product of Z27 and Z3
M81 (semidirect product of Z27 and Z3) 6 2 element structure of M81
Wreath product of Z3 and Z3 7 3 element structure of wreath product of Z3 and Z3
SmallGroup(81,8) 8 3 element structure of SmallGroup(81,8)
SmallGroup(81,9) 9 3 element structure of SmallGroup(81,9)
SmallGroup(81,10) 10 3 element structure of SmallGroup(81,10)
Direct product of Z9 and E9 11 1 element structure of direct product of Z9 and E9
Direct product of prime-cube order group:U(3,3) and Z3 12 2 element structure of direct product of prime-cube order group:U(3,3) and Z3
Direct product of semidirect product of Z9 and Z3 and Z3 13 2 element structure of direct product of semidirect product of Z9 and Z3 and Z3
Central product of prime-cube order group:U(3,3) and Z9 14 2 element structure of central product of prime-cube order group:U(3,3) and Z9
Elementary abelian group:E81 15 1 element structure of elementary abelian group:E81


1-isomorphism

Pairs where one of the groups is abelian

Of the 15 groups of order 81, 5 are abelian, 6 have nilpotency class two, and 4 have nilpotency class three. Via the Baer correspondence, each of the groups of class two has a Baer Lie ring, and in particular is 1-isomorphic to the additive group of that Lie ring. Of the 4 groups of nilpotency class three, only one (SmallGroup(81,8)) is 1-isomorphic to an abelian group. There are no 1-isomorphisms between pairs where both members are non-abelian.

Non-abelian member of pair Nilpotency class GAP ID Abelian member of pair GAP ID Nature of the 1-isomorphism Description of the 1-isomorphism Best perspective Alternative perspective
M81 2 6 direct product of Z27 and Z3 5 Baer correspondence Baer correspondence between M81 and its Lie ring
nontrivial semidirect product of Z9 and Z9 2 4 direct product of Z9 and Z9 2 Baer correspondence Baer correspondence between nontriival semidirect product of Z9 and Z9 and its Lie ring
SmallGroup(81,3) 2 3 direct product of Z9 and E9 11 Baer correspondence Baer correspondence between SmallGroup(81,3) and its Lie ring
direct product of semidirect product of Z9 and Z3 and Z3 2 13 direct product of Z9 and E9 11 Baer correspondence Baer correspondence between direct product of semidirect product of Z9 and Z3 and Z3 and its Lie ring
central product of prime-cube order group:U(3,3) and Z9 2 14 direct product of Z9 and E9 11 Baer correspondence Baer correspondence between central product of prime-cube order group:U(3,3) and Z9 and its Lie ring
direct product of prime-cube order group:U(3,3) and Z3 2 12 elementary abelian group:E81 15 Baer correspondence Baer correspondence between direct product of prime-cube order group:U(3,3) and Z3 and its Lie ring
SmallGroup(81,8) 3 8 direct product of Z9 and E9 11 ?

Grouping by abelian member

Abelian member GAP ID second part Total number of members (including abelian member) Other members GAP IDs second part (in order of listing)
direct product of Z9 and Z9 2 2 nontrivial semidirect product of Z9 and Z9 4
direct product of Z27 and Z3 5 2 M81 6
direct product of Z9 and E9 11 5 SmallGroup(81,3), SmallGroup(81,8), direct product of semidirect product of Z9 and Z3 and Z3, central product of prime-cube order group:U(3,3) and Z9 3, 8, 13, 14
elementary abelian group:E81 15 2 direct product of prime-cube order group:U(3,3) and Z3 12

Groupings that do not have any abelian members

There are no 1-isomorphisms between pairs where both members are non-abelian for order 81. Thus, there are no such groupings.

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

Order statistics raw data

Group Second part of GAP ID Number of elements of order 1 Number of elements of order 3 Number of elements of order 9 Number of elements of order 27 Number of elements of order 81
Cyclic group:Z81 1 1 2 6 18 54
Direct product of Z9 and Z9 2 1 8 72 0 0
SmallGroup(81,3) 3 1 26 54 0 0
Nontrivial semidirect product of Z9 and Z9 4 1 8 72 0 0
Direct product of Z27 and Z3 5 1 8 18 54 0
M81 6 1 8 18 54 0
Wreath product of Z3 and Z3 7 1 44 36 0 0
SmallGroup(81,8) 8 1 26 54 0 0
SmallGroup(81,9) 9 1 62 18 0 0
SmallGroup(81,10) 10 1 8 72 0 0
Direct product of Z9 and E9 11 1 26 54 0 0
Direct product of prime-cube order group:U(3,3) and Z3 12 1 80 0 0 0
Direct product of semidirect product of Z9 and Z3 and Z3 13 1 26 54 0 0
SmallGroup(81,14) 14 1 26 54 0 0
Elementary abelian group:E81 15 1 80 0 0 0

Here is the GAP code to generate these order statistics:[SHOW MORE]

Here are the cumulative order statistics.

Group Second part of GAP ID Number of 1st roots Number of 3rd roots Number of 9th roots Number of 27th roots Number of 81th roots
Cyclic group:Z81 1 1 3 9 27 81
Direct product of Z9 and Z9 2 1 9 81 81 81
SmallGroup(81,3) 3 1 27 81 81 81
Nontrivial semidirect product of Z9 and Z9 4 1 9 81 81 81
Direct product of Z27 and Z3 5 1 9 27 81 81
M81 6 1 9 27 81 81
Wreath product of Z3 and Z3 7 1 45 81 81 81
SmallGroup(81,8) 8 1 27 81 81 81
SmallGroup(81,9) 9 1 63 81 81 81
SmallGroup(81,10) 10 1 9 81 81 81
Direct product of Z9 and E9 11 1 27 81 81 81
Direct product of prime-cube order group:U(3,3) and Z3 12 1 81 81 81 81
Direct product of semidirect product of Z9 and Z3 and Z3 13 1 27 81 81 81
SmallGroup(81,14) 14 1 27 81 81 81
Elementary abelian group:E81 15 1 81 81 81 81

Here is the GAP code to generate these cumulative order statistics:[SHOW MORE]

Equivalence classes based on order statistics

Here, we discuss the equivalence classes of groups of order 81 up to being order statistics-equivalent finite groups and up to the stronger notion of being 1-isomorphic groups (which means there is a bijection that restricts to isomorphisms on cyclic subgroups). See also order statistics-equivalent not implies 1-isomorphic.

Order statistics Order statistics (cumulative) Number of groups Number of equivalence classes up to 1-isomorphism Members of first equivalence class Members of second equivalence class Abelian group with these order statistics? Cumulative order statistics all powers of 3?
1,2,6,18,54 1,3,9,27,81 1 1 Cyclic group:Z81 (ID:1) -- Yes Yes
1,8,18,54,0 1,9,27,81,81 2 1 Direct product of Z27 and Z3 (ID:5) and M81 (ID:6) -- Yes Yes
1,8,72,0,0 1,9,81,81,81 3 2 Direct product of Z9 and Z9 (ID:2), Semidirect product of Z9 and Z9 (ID:4) SmallGroup(81,10) (ID:10) Yes Yes
1,26,54,0,0 1,27,81,81,81 5 1 or 2 Direct product of Z9 and E9 (ID:11), SmallGroup(81,3) (ID:3), Direct product of semidirect product of Z9 and Z3 and Z3 (ID:13), Central product of prime-cube order group:U(3,3) and Z9 (ID:14), SmallGroup(81,8) (ID:8) Yes Yes
1,44,36,0,0 1,45,81,81,81 1 1 Wreath product of Z3 and Z3 (ID:7) -- No No
1,62,18,0,0 1,63,81,81,81 1 1 SmallGroup(81,9) (ID:9) -- No No
1,80,0,0,0 1,81,81,81,81 2 1 Elementary abelian group:E81 (ID:15) and Direct product of prime-cube order group:U(3,3) and Z3 (ID:12) -- Yes Yes
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