Element structure of groups of prime-fourth order

This article gives specific information, namely, element structure, about a family of groups, namely: groups of prime-fourth order.
View element structure of group families | View other specific information about groups of prime-fourth order

Particular cases

Value of prime $p$	Value of $p^4$	Information on groups of order $p^4$	Information on element structure of groups of order $p^4$	Anomalous?
2	16	groups of order 16	element structure of groups of order 16	Highly
3	81	groups of order 81	element structure of groups of order 81	Somewhat
5	625	groups of order 625	element structure of groups of order 625	No
7	2401	groups of order 2401	element structure of groups of order 2401	No
11	14641	groups of order 14641	element structure of groups of order 14641	No

The prime $p = 2$ and $p = 3$ behave somewhat differently from the other primes.

Conjugacy class sizes

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

Grouping by conjugacy class sizes

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

In particular, we have:

Nilpotency class	Number of conjugacy classes of size 1	Number of conjugacy classes of size $p$	Number of conjugacy classes of size $p^2$	Total number of conjugacy classes	Number of groups case $p = 2$	Number of groups case $p \ge 3$
1	$p^4$	0	0	$p^4$	5	5
2	$p^2$	$p^3 - p$	0	$p^3 + p^2 - p$	6	6
3	$p$	$p^2 - 1$	$p^2 - p$	$2p^2 - 1$	3	4

Grouping by cumulative conjugacy class sizes (number of elements)

Nilpotency class	Number of elements conjugacy classes of size 1	Number of elements in conjugacy classes of size dividing $p$	Number of elements in conjugacy classes of size dividing $p^2$	Total number of conjugacy classes	Number of groups case $p = 2$	Number of groups case $p \ge 3$
1	$p^4$	$p^4$	$p^4$	$p^4$	5	5
2	$p^2$	$p^4$	$p^4$	$p^3 + p^2 - p$	6	6
3	$p$	$p^3$	$p^4$	$2p^2 - 1$	3	4

Note that it is true in this case that the number of elements in conjugacy classes of size dividing any number itself divides the order of the group (in particular, all these numbers are powers of $p$). However, this is not true for all groups and in fact an analogous statement fails for groups of prime-sixth order (see element structure of groups of prime-sixth order). For more, see:

• All cumulative conjugacy class size statistics values divide the order of the group for groups up to prime-fifth order
• There exist groups of prime-sixth order in which the cumulative conjugacy class size statistics values do not divide the order of the group

Correspondence between conjugacy class sizes and degrees of irreducible representations

For groups of order $p^4$, it is also true that the nilpotency class determines the degrees of irreducible representations in terms of the underlying prime. Further information: Nilpotency class and order determine conjugacy class size statistics for groups up to prime-fourth order

In fact, the nilpotency class, conjugacy class size statistics, and degrees of irreducible representations all determine each other. Knowing any one of these determines the other two once we fix the underlying prime. Below is the correspondence:

Nilpotency class	Number of conjugacy classes of size 1	Number of conjugacy classes of size $p$	Number of conjugacy classes of size $p^2$	Total number of conjugacy classes = number of irreducible representations	Number of irreps of degree 1	Number of irreps of degree $p$
1	$p^4$	0	0	$p^4$	$p^4$	0
2	$p^2$	$p^3 - p$	0	$p^3 + p^2 - p$	$p^3$	$p^2 - p$
3	$p$	$p^2 - 1$	$p^2 - p$	$2p^2 - 1$	$p^2$	$p^2 - 1$

Note that the phenomenon of the conjugacy class size statistics and degrees of irreducible representations determining one another is not true for all orders:

• Degrees of irreducible representations need not determine conjugacy class size statistics
• Conjugacy class size statistics need not determine degrees of irreducible representations

1-isomorphism

We say that two groups are 1-isomorphic groups if there is a bijection between them that restricts to an isomorphism on every cyclic subgroup of either side.

Case $p$ at least 5

In this case, all groups of order $p^4$ are 1-isomorphic to abelian groups via the Lazard correspondence. This is because all the groups have nilpotency class at most three, which is less than the underlying prime, and hence the Lazard correspondence applies.

Further, the nature of the Lazard correspondence is similar for all choices of $p$. More detailed information is presented in the table below:

Partition of $4$	Corresponding abelian group (generic prime $p$)	GAP ID (second part)	List of GAP IDs of class two groups that are 1-isomorphic to it via the Baer correspondence	Number of such class two groups	List of GAP IDs of class three (not class two) groups that are 1-isomorphic to it via the Lazard correspondence	Number of such class three groups
4	cyclic group of prime-fourth order	1	--	0	--	0
2 + 2	direct product of cyclic group of prime-square order and cyclic group of prime-square order	2	4	1	--	0
3 + 1	direct product of cyclic group of prime-cube order and cyclic group of prime order	5	6	1	--	0
2 + 1 + 1	direct product of cyclic group of prime-square order and elementary abelian group of prime-square order	11	3, 13, 14	3	8, 9, 10	3
1 + 1 + 1 + 1	elementary abelian group of prime-fourth order	15	12	1	7	1
Total	5	--	--	6	--	4

Case $p = 3$

Further information: element structure of groups of order 81

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
semidirect product of Z27 and Z3	2	6	direct product of Z27 and Z3	5	Baer correspondence	Baer correspondence between semidirect product of Z27 and Z3 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	?

Case $p = 2$

Further information: Element structure of groups of order 16#1-isomorphism

Of the 14 groups of order 16, 10 are not 1-isomorphic to any other group. The remaining four come in pairs. These two pairs are described below.

Non-abelian member of pair	GAP ID	Abelian member of pair	GAP ID	The 1-isomorphism arises as a ...	Description of the 1-isomorphism	Best perspective	Alternative perspective
central product of D8 and Z4	13	direct product of Z4 and V4	10	cocycle halving generalization of Baer correspondence, the intermediate object being a class two Lie cring	generalized Baer correspondence between central product of D8 and Z4 and direct product of Z4 and V4	second cohomology group for trivial group action of V4 on Z4#Generalized Baer Lie rings	second cohomology group for trivial group action of E8 on Z2#Direct sum decomposition
M16	6	direct product of Z8 and Z2	5	cocycle halving generalization of Baer correspondence, the intermediate object being a class two Lie cring	generalized Baer correspondence between M16 and direct product of Z8 and Z2	second cohomology group for trivial group action of V4 on Z4#Generalized Baer Lie rings	second cohomology group for trivial group action of direct product of Z4 and Z2 on Z2#Generalized Baer Lie rings

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

Case $p$ at least 5

For groups of order $p^4$ with $p \ge 5$, the equivalence classes of groups based on order statistics coincide precisely with the equivalence classes based on 1-isomorphism, and every equivalence class contains exactly one abelian group (see the section #1-isomorphism). The convention for order statistics is as follows: the comma-separated values give the number of elements of order 1, $p$, $p^2$, $p^3$, and $p^4$ respectively. For cumulative order statistics, these are the number of elements of order dividing 1, $p$, $p^2$, $p^3$, and $p^4$ respectively::

Order statistics	Order statistics (cumulative)	Partition for abelian group	Corresponding abelian group (generic prime $p$)	GAP ID (second part)	List of GAP IDs of class two groups that are order statistics-equivalent to it	Number of such class two groups	List of GAP IDs of class three (not class two) groups that are order statistics-equivalen to it	Number of such class three groups
1,$p - 1$, $p^2 - p$, $p^3 - p^2$, $p^4 - p^3$	1, $p$, $p^2$, $p^3$, $p^4$	4	cyclic group of prime-fourth order	1	--	0	--	0
1, $p^2 - 1$, $p^4 - p^2$, 0, 0	1, $p^2$, $p^4$, $p^4$, $p^4$	2 + 2	direct product of cyclic group of prime-square order and cyclic group of prime-square order	2	4	1	--	0
1, $p^2 - 1$, $p^3 - p^2$, $p^4 - p^3$, 0	1, $p^2$, $p^3$, $p^4$, $p^4$	3 + 1	direct product of cyclic group of prime-cube order and cyclic group of prime order	5	6	1	--	0
1, $p^3 - 1$, $p^4 - p^3$, 0, 0	1, $p^3$, $p^4$, $p^4$, $p^4$	2 + 1 + 1	direct product of cyclic group of prime-square order and elementary abelian group of prime-square order	11	3, 13, 14	3	8, 9, 10	3
1, $p^4 - 1$, 0, 0, 0	1, $p^4$, $p^4$, $p^4$, $p^4$	1 + 1 + 1 + 1	elementary abelian group of prime-fourth order	15	12	1	7	1
Total	5	--	--	6	--	4

Case $p = 3$

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 Semidirect product of Z27 and Z3 (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

Case $p = 2$

Further information: Element structure of groups of order 16#1-isomorphism

Note that for groups of order $2^4 = 16$, not all groups are order statistics-equivalent to an abelian group, and even for those that are, the group need not be 1-isomorphic to an abelian group. Here, we discuss the equivalence classes of groups of order 16 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	Members of third equivalence class	Abelian group with these order statistics?	Cumulative order statistics all powers of 2?
1,1,2,4,8	1,2,4,8,16	1	1	Cyclic group:Z16 (ID:1)			Yes	Yes
1,1,10,4,0	1,2,12,16,16	1	1	Generalized quaternion group:Q16 (ID:9)			No	No
1,3,4,8,0	1,4,8,16,16	2	1	Direct product of Z8 and Z2 (ID:5) and M16 (ID:6)			Yes	Yes
1,3,12,0,0	1,4,16,16,16	3	3	Direct product of Z4 and Z4 (ID:2) (characterized by having three squares of order 2)	Nontrivial semidirect product of Z4 and Z4 (ID:4) (characterized by having two squares of order 2)	Direct product of Q8 and Z2 (ID:12)	Yes	Yes
1,5,6,4,0	1,6,12,16,16	1	1	Semidihedral group:SD16 (ID:8)			No	No
1,7,8,0,0	1,8,16,16,16	3	2	Direct product of Z4 and V4 (ID:10) and Central product of D8 and Z4 (ID:13) (characterized by having exactly one square of order 2)	SmallGroup(16,3) (ID:3)(characterized by having two squares of order 2)		Yes	Yes
1,9,2,4,0	1,10,12,16,16	1	1	Dihedral group:D16 (ID:7)			No	No
1,11,4,0,0	1,12,16,16,16	1	1	Direct product of D8 and Z2 (ID:11)			No	No
1,15,0,0,0	1,16,16,16,16	1	1	Elementary abelian group:E16 (ID:14)			Yes	Yes

Here are the GAP commands to produce a list sorted by order statistics: [SHOW MORE]

gap> F := List(AllSmallGroups(16),G -> List(Set(G),Order));;
gap> K := List(F,L->[Length(Filtered(L,x -> x = 1)),
> Length(Filtered(L,x -> x = 2)),Length(Filtered(L,x -> x = 4)),
> Length(Filtered(L,x -> x = 8)),Length(Filtered(L,x -> x = 16))]);;
gap> M := List([1..14], i ->[K[i],i]);;
gap> S := SortedList(M);

Here is GAP's output (note that this does not distinguish based on 1-isomorphism):

[ [ [ 1, 1, 2, 4, 8 ], 1 ], [ [ 1, 1, 10, 4, 0 ], 9 ], [ [ 1, 3, 4, 8, 0 ], 5 ], [ [ 1, 3, 4, 8, 0 ], 6 ], [ [ 1, 3, 12, 0, 0 ], 2 ],
  [ [ 1, 3, 12, 0, 0 ], 4 ], [ [ 1, 3, 12, 0, 0 ], 12 ], [ [ 1, 5, 6, 4, 0 ], 8 ], [ [ 1, 7, 8, 0, 0 ], 3 ], [ [ 1, 7, 8, 0, 0 ], 10 ],
  [ [ 1, 7, 8, 0, 0 ], 13 ], [ [ 1, 9, 2, 4, 0 ], 7 ], [ [ 1, 11, 4, 0, 0 ], 11 ], [ [ 1, 15, 0, 0, 0 ], 14 ] ]