Element structure of groups of order 16

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

Note that because number of nth roots is a multiple of n, we see that the number of elements whose order is $1$ or $2$ is odd, while all the other numbers are even. The total number of $n^{th}$ roots is even for all $n=2^{k},k\geq 1$.

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	Number of elements of order 16
Cyclic group:Z16	1	5	1	1	2	4	8
Direct product of Z4 and Z4	2	3	1	3	12	0	0
SmallGroup(16,3)	3	9	1	7	8	0	0
Nontrivial semidirect product of Z4 and Z4	4	10	1	3	12	0	0
Direct product of Z8 and Z2	5	4	1	3	4	8	0
M16	6	11	1	3	4	8	0
Dihedral group:D16	7	12	1	9	2	4	0
Semidihedral group:SD16	8	13	1	5	6	4	0
Generalized quaternion group:Q16	9	14	1	1	10	4	0
Direct product of Z4 and V4	10	2	1	7	8	0	0
Direct product of D8 and Z2	11	6	1	11	4	0	0
Direct product of Q8 and Z2	12	7	1	3	12	0	0
Central product of D8 and Z4	13	8	1	7	8	0	0
Elementary abelian group:E16	14	1	1	15	0	0	0

Here are the GAP commands to generate these 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 ->[i,K[i]]);

GAP's output is:

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

Here are the cumulative statistics, which gives the number of $n^{th}$ roots:

Group	Second part of GAP ID	Hall-Senior number	Number of 1st roots	Number of 2nd roots	Number of 4th roots	Number of 8th rots	Number of 16th roots
Cyclic group:Z16	1	5	1	2	4	8	16
Direct product of Z4 and Z4	2	3	1	4	16	16	16
SmallGroup(16,3)	3	9	1	8	16	16	16
Nontrivial semidirect product of Z4 and Z4	4	10	1	4	16	16	16
Direct product of Z8 and Z2	5	4	1	4	8	16	16
M16	6	11	1	4	8	16	16
Dihedral group:D16	7	12	1	10	12	16	16
Semidihedral group:SD16	8	13	1	6	12	16	16
Generalized quaternion group:Q16	9	14	1	2	12	16	16
Direct product of Z4 and V4	10	2	1	8	16	16	16
Direct product of D8 and Z2	11	6	1	12	16	16	16
Direct product of Q8 and Z2	12	7	1	4	16	16	16
Central product of D8 and Z4	13	8	1	8	16	16	16
Elementary abelian group:E16	14	1	1	16	16	16	16

Here are the GAP commands to generate these cumulative statistics: [SHOW MORE]

gap> F := List(AllSmallGroups(16),G -> List(Set(G),Order));;
gap> J := 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> N := List([1..14], i ->[i,J[i]]);

Here is GAP's output:

[ [ 1, [ 1, 2, 4, 8, 16 ] ], [ 2, [ 1, 4, 16, 16, 16 ] ], [ 3, [ 1, 8, 16, 16, 16 ] ], [ 4, [ 1, 4, 16, 16, 16 ] ], [ 5, [ 1, 4, 8, 16, 16 ] ],
  [ 6, [ 1, 4, 8, 16, 16 ] ], [ 7, [ 1, 10, 12, 16, 16 ] ], [ 8, [ 1, 6, 12, 16, 16 ] ], [ 9, [ 1, 2, 12, 16, 16 ] ], [ 10, [ 1, 8, 16, 16, 16 ] ],
  [ 11, [ 1, 12, 16, 16, 16 ] ], [ 12, [ 1, 4, 16, 16, 16 ] ], [ 13, [ 1, 8, 16, 16, 16 ] ], [ 14, [ 1, 16, 16, 16, 16 ] ] ]

Group properties based on order statistics

Group	GAP ID	Hall-Senior number	abelian group of prime power order	group of prime power order admitting a bijective quasihomomorphism to an abelian group	group of prime power order 1-isomorphic to an abelian group	group of prime power order order statistics-equivalent to an abelian group	finite p-group in which the number of nth roots is a power of p for all n	order statistics-unique finite group
Cyclic group:Z16	1	5	Yes	Yes	Yes	Yes	Yes	Yes
Direct product of Z4 and Z4	2	3	Yes	Yes	Yes	Yes	Yes	No
SmallGroup(16,3)	3	9	No	No	No	Yes	Yes	No
Nontrivial semidirect product of Z4 and Z4	4	10	No	No	No	Yes	Yes	No
Direct product of Z8 and Z2	5	4	Yes	Yes	Yes	Yes	Yes	No
M16	6	11	No	Yes	Yes	Yes	Yes	No
Dihedral group:D16	7	12	No	No	No	No	No	Yes
Semidihedral group:SD16	8	13	No	No	No	No	No	Yes
Generalized quaternion group:Q16	9	14	No	No	No	No	No	Yes
Direct product of Z4 and V4	10	2	Yes	Yes	Yes	Yes	Yes	No
Direct product of D8 and Z2	11	6	No	No	No	No	No	Yes
Direct product of Q8 and Z2	12	7	No	No	No	Yes	Yes	No
Central product of D8 and Z4	13	8	No	?	Yes	Yes	Yes	No
Elementary abelian group:E16	14	1	Yes	Yes	Yes	Yes	Yes	Yes

Equivalence classes based on order statistics

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 ] ]

Conjugacy class-cum-order statistics

Because of the small order, it turns out that the nilpotency class completely determines the number of conjugacy classes of each size.

Nilpotency class one: abelian groups

There are 16 conjugacy classes of size 1. See the order statistics section for the order statistics of these groups.

Nilpotency class exactly two

There are 4 conjugacy classes of size 1, comprising the central elements, and 6 conjugacy classes of size two. Listed below are the conjugacy classes sorted by order. Note that the numbers listed are the numbers of conjugacy classes with the given size/order specifications. To obtain the number of elements, multiply by the size of the conjugacy class.

Group	Second part of GAP ID	Hall-Senior number	size 1, order 1	size 1, order 2	size 1, order 4	size 2, order 2	size 2, order 4	size 2, order 8
SmallGroup(16,3)	3	9	1	3	0	2	4	0
Nontrivial semidirect product of Z4 and Z4	4	10	1	3	0	0	6	0
M16	6	11	1	1	2	1	1	4
Direct product of D8 and Z2	11	6	1	3	0	4	2	0
Direct product of Q8 and Z2	12	7	1	3	0	0	6	0
Central product of D8 and Z4	13	8	1	1	2	3	3	0

Nilpotency class exactly three

There are 2 conjugacy classes of size 1, comprising the central elements, 3 conjugacy classes of size 2, and 2 conjugacy classes of size 4. Listed below are the conjugacy classes sorted by order. Note that the numbers listed are the numbers of conjugacy classes with the given size/order specifications. To obtain the number of elements, multiply by the size of the conjugacy class.

Group	Second part of GAP ID	Hall-Senior number	size 1, order 1	size 1, order 2	size 2, order 2	size 2, order 4	size 2, order 8	size 4, order 2	size 4, order 4
Dihedral group:D16	7	12	1	1	0	1	2	2	0
Semidihedral group:SD16	8	13	1	1	0	1	2	1	1
Generalized quaternion group:Q16	9	14	1	1	0	1	2	0	2