How many groups are there of each order up to 16, and what are their multiplication tables?
combinatorics/finite.groups New
New entry, edited from material collected for the archive and not yet in a released version.
Problem
How many groups are there of each order up to 16, and what are their multiplication tables? [ Editor: problem statement reconstructed from the postings. ]
Show solution
I've written a program to generate finite groups, and I'm posting the
results here for the benefit of the curious. Barring bugs in my program
and editing errors, the attached list should include all groups of order
16 or less. For best results, the list should be printed on a device
that allows underlining.
For those of you who don't know what a group is, a group consists of a
set S and a binary operator + such that:
1) For all a and b in S, a + b is in S.
2) There is an element 0 in S such that a + 0 = 0 + a = a for all a.
3) For all a there is a b such that a + b = b + a = 0.
4) For all a, b, and c, (a + b) + c = a + (b + c).
The order of a group is simply the number of elements in the set S.
---------------------------------
order 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
num. groups 1 1 1 2 1 2 1 5 2 2 1 6 1 2 1 19
non-commutative 1 2 1 4 1 14
[ Editor: the counts of 6 groups of order 12 and 19 of order 16 are
corrected to 5 and 14 in the follow-ups at the end of this entry. ]
1 group of order 1:
___
A|A
1 group of order 2:
_____
A|A B
B|B A
1 group of order 3:
_______
A|A B C
B|B C A
C|C A B
2 groups of order 4:
_________ _________
A|A B C D A|A B C D
B|B A D C B|B C D A
C|C D A B C|C D A B
D|D C B A D|D A B C
1 group of order 5:
___________
A|A B C D E
B|B C D E A
C|C D E A B
D|D E A B C
E|E A B C D
2 groups of order 6:
_____________ _____________
A|A B C D E F A|A B C D E F
B|B C A E F D B|B C D E F A
C|C A B F D E C|C D E F A B
D|D F E A C B D|D E F A B C
E|E D F B A C E|E F A B C D
F|F E D C B A F|F A B C D E
1 group of order 7:
_______________
A|A B C D E F G
B|B C D E F G A
C|C D E F G A B
D|D E F G A B C
E|E F G A B C D
F|F G A B C D E
G|G A B C D E F
5 groups of order 8:
_________________ _________________ _________________
A|A B C D E F G H A|A B C D E F G H A|A B C D E F G H
B|B A D C F E H G B|B C D A F G H E B|B C D A F G H E
C|C D A B G H E F C|C D A B G H E F C|C D A B G H E F
D|D C B A H G F E D|D A B C H E F G D|D A B C H E F G
E|E F G H A B C D E|E F G H A B C D E|E H G F A D C B
F|F E H G B A D C F|F G H E B C D A F|F E H G B A D C
G|G H E F C D A B G|G H E F C D A B G|G F E H C B A D
H|H G F E D C B A H|H E F G D A B C H|H G F E D C B A
_________________ _________________
A|A B C D E F G H A|A B C D E F G H
B|B C D A F G H E B|B C D E F G H A
C|C D A B G H E F C|C D E F G H A B
D|D A B C H E F G D|D E F G H A B C
E|E H G F C B A D E|E F G H A B C D
F|F E H G D C B A F|F G H A B C D E
G|G F E H A D C B G|G H A B C D E F
H|H G F E B A D C H|H A B C D E F G
2 groups of order 9:
___________________ ___________________
A|A B C D E F G H I A|A B C D E F G H I
B|B C A E F D H I G B|B C D E F G H I A
C|C A B F D E I G H C|C D E F G H I A B
D|D E F G H I A B C D|D E F G H I A B C
E|E F D H I G B C A E|E F G H I A B C D
F|F D E I G H C A B F|F G H I A B C D E
G|G H I A B C D E F G|G H I A B C D E F
H|H I G B C A E F D H|H I A B C D E F G
I|I G H C A B F D E I|I A B C D E F G H
2 groups of order 10:
_____________________ _____________________
A|A B C D E F G H I J A|A B C D E F G H I J
B|B C D E A G H I J F B|B C D E F G H I J A
C|C D E A B H I J F G C|C D E F G H I J A B
D|D E A B C I J F G H D|D E F G H I J A B C
E|E A B C D J F G H I E|E F G H I J A B C D
F|F J I H G A E D C B F|F G H I J A B C D E
G|G F J I H B A E D C G|G H I J A B C D E F
H|H G F J I C B A E D H|H I J A B C D E F G
I|I H G F J D C B A E I|I J A B C D E F G H
J|J I H G F E D C B A J|J A B C D E F G H I
1 group of order 11:
_______________________
A|A B C D E F G H I J K
B|B C D E F G H I J K A
C|C D E F G H I J K A B
D|D E F G H I J K A B C
E|E F G H I J K A B C D
F|F G H I J K A B C D E
G|G H I J K A B C D E F
H|H I J K A B C D E F G
I|I J K A B C D E F G H
J|J K A B C D E F G H I
K|K A B C D E F G H I J
6 groups of order 12:
_________________________ _________________________
A|A B C D E F G H I J K L A|A B C D E F G H I J K L
B|B C A E F D H I G K L J B|B C A E F D H I G K L J
C|C A B F D E I G H L J K C|C A B F D E I G H L J K
D|D G J A L H B F K C I E D|D I L G C K A F J B H E
E|E H K B J I C D L A G F E|E G J H A L B D K C I F
F|F I L C K G A E J B H D F|F H K I B J C E L A G D
G|G J D L H A F K B I E C G|G J E A L H D K B I F C
H|H K E J I B D L C G F A H|H K F B J I E L C G D A
I|I L F K G C E J A H D B I|I L D C K G F J A H E B
J|J D G H A L K B F E C I J|J E G L H A K B D F C I
K|K E H I B J L C D F A G K|K F H J I B L C E D A G
L|L F I G C K J A E D B H L|L D I K G C J A F E B H
_________________________ _________________________
A|A B C D E F G H I J K L A|A B C D E F G H I J K L
B|B C D E F A H I J K L G B|B C D E F A H I J K L G
C|C D E F A B I J K L G H C|C D E F A B I J K L G H
D|D E F A B C J K L G H I D|D E F A B C J K L G H I
E|E F A B C D K L G H I J E|E F A B C D K L G H I J
F|F A B C D E L G H I J K F|F A B C D E L G H I J K
G|G H I J K L A B C D E F G|G L K J I H A F E D C B
H|H I J K L G B C D E F A H|H G L K J I B A F E D C
I|I J K L G H C D E F A B I|I H G L K J C B A F E D
J|J K L G H I D E F A B C J|J I H G L K D C B A F E
K|K L G H I J E F A B C D K|K J I H G L E D C B A F
L|L G H I J K F A B C D E L|L K J I H G F E D C B A
_________________________ _________________________
A|A B C D E F G H I J K L A|A B C D E F G H I J K L
B|B C D E F A H I J K L G B|B C D E F G H I J K L A
C|C D E F A B I J K L G H C|C D E F G H I J K L A B
D|D E F A B C J K L G H I D|D E F G H I J K L A B C
E|E F A B C D K L G H I J E|E F G H I J K L A B C D
F|F A B C D E L G H I J K F|F G H I J K L A B C D E
G|G L K J I H D C B A F E G|G H I J K L A B C D E F
H|H G L K J I E D C B A F H|H I J K L A B C D E F G
I|I H G L K J F E D C B A I|I J K L A B C D E F G H
J|J I H G L K A F E D C B J|J K L A B C D E F G H I
K|K J I H G L B A F E D C K|K L A B C D E F G H I J
L|L K J I H G C B A F E D L|L A B C D E F G H I J K
1 group of order 13:
___________________________
A|A B C D E F G H I J K L M
B|B C D E F G H I J K L M A
C|C D E F G H I J K L M A B
D|D E F G H I J K L M A B C
E|E F G H I J K L M A B C D
F|F G H I J K L M A B C D E
G|G H I J K L M A B C D E F
H|H I J K L M A B C D E F G
I|I J K L M A B C D E F G H
J|J K L M A B C D E F G H I
K|K L M A B C D E F G H I J
L|L M A B C D E F G H I J K
M|M A B C D E F G H I J K L
2 groups of order 14:
_____________________________ _____________________________
A|A B C D E F G H I J K L M N A|A B C D E F G H I J K L M N
B|B C D E F G A I J K L M N H B|B C D E F G H I J K L M N A
C|C D E F G A B J K L M N H I C|C D E F G H I J K L M N A B
D|D E F G A B C K L M N H I J D|D E F G H I J K L M N A B C
E|E F G A B C D L M N H I J K E|E F G H I J K L M N A B C D
F|F G A B C D E M N H I J K L F|F G H I J K L M N A B C D E
G|G A B C D E F N H I J K L M G|G H I J K L M N A B C D E F
H|H N M L K J I A G F E D C B H|H I J K L M N A B C D E F G
I|I H N M L K J B A G F E D C I|I J K L M N A B C D E F G H
J|J I H N M L K C B A G F E D J|J K L M N A B C D E F G H I
K|K J I H N M L D C B A G F E K|K L M N A B C D E F G H I J
L|L K J I H N M E D C B A G F L|L M N A B C D E F G H I J K
M|M L K J I H N F E D C B A G M|M N A B C D E F G H I J K L
N|N M L K J I H G F E D C B A N|N A B C D E F G H I J K L M
1 group of order 15:
_______________________________
A|A B C D E F G H I J K L M N O
B|B C D E F G H I J K L M N O A
C|C D E F G H I J K L M N O A B
D|D E F G H I J K L M N O A B C
E|E F G H I J K L M N O A B C D
F|F G H I J K L M N O A B C D E
G|G H I J K L M N O A B C D E F
H|H I J K L M N O A B C D E F G
I|I J K L M N O A B C D E F G H
J|J K L M N O A B C D E F G H I
K|K L M N O A B C D E F G H I J
L|L M N O A B C D E F G H I J K
M|M N O A B C D E F G H I J K L
N|N O A B C D E F G H I J K L M
O|O A B C D E F G H I J K L M N
19 groups of order 16:
_________________________________ _________________________________
A|A B C D E F G H I J K L M N O P A|A B C D E F G H I J K L M N O P
B|B A D C F E H G J I L K N M P O B|B C D A F G H E J K L I N O P M
C|C D A B G H E F K L I J O P M N C|C D A B G H E F K L I J O P M N
D|D C B A H G F E L K J I P O N M D|D A B C H E F G L I J K P M N O
E|E F G H A B C D M N O P I J K L E|E F G H A B C D M N O P I J K L
F|F E H G B A D C N M P O J I L K F|F G H E B C D A N O P M J K L I
G|G H E F C D A B O P M N K L I J G|G H E F C D A B O P M N K L I J
H|H G F E D C B A P O N M L K J I H|H E F G D A B C P M N O L I J K
I|I J K L M N O P A B C D E F G H I|I J K L M N O P A B C D E F G H
J|J I L K N M P O B A D C F E H G J|J K L I N O P M B C D A F G H E
K|K L I J O P M N C D A B G H E F K|K L I J O P M N C D A B G H E F
L|L K J I P O N M D C B A H G F E L|L I J K P M N O D A B C H E F G
M|M N O P I J K L E F G H A B C D M|M N O P I J K L E F G H A B C D
N|N M P O J I L K F E H G B A D C N|N O P M J K L I F G H E B C D A
O|O P M N K L I J G H E F C D A B O|O P M N K L I J G H E F C D A B
P|P O N M L K J I H G F E D C B A P|P M N O L I J K H E F G D A B C
_________________________________ _________________________________
A|A B C D E F G H I J K L M N O P A|A B C D E F G H I J K L M N O P
B|B C D A F G H E J K L I N O P M B|B C D A F G H E J K L I N O P M
C|C D A B G H E F K L I J O P M N C|C D A B G H E F K L I J O P M N
D|D A B C H E F G L I J K P M N O D|D A B C H E F G L I J K P M N O
E|E F G H A B C D M N O P I J K L E|E F G H A B C D M N O P I J K L
F|F G H E B C D A N O P M J K L I F|F G H E B C D A N O P M J K L I
G|G H E F C D A B O P M N K L I J G|G H E F C D A B O P M N K L I J
H|H E F G D A B C P M N O L I J K H|H E F G D A B C P M N O L I J K
I|I J K L O P M N A B C D G H E F I|I L K J M P O N A D C B E H G F
J|J K L I P M N O B C D A H E F G J|J I L K N M P O B A D C F E H G
K|K L I J M N O P C D A B E F G H K|K J I L O N M P C B A D G F E H
L|L I J K N O P M D A B C F G H E L|L K J I P O N M D C B A H G F E
M|M N O P K L I J E F G H C D A B M|M P O N I L K J E H G F A D C B
N|N O P M L I J K F G H E D A B C N|N M P O J I L K F E H G B A D C
O|O P M N I J K L G H E F A B C D O|O N M P K J I L G F E H C B A D
P|P M N O J K L I H E F G B C D A P|P O N M L K J I H G F E D C B A
_________________________________ _________________________________
A|A B C D E F G H I J K L M N O P A|A B C D E F G H I J K L M N O P
B|B C D A F G H E J K L I N O P M B|B C D A F G H E J K L I N O P M
C|C D A B G H E F K L I J O P M N C|C D A B G H E F K L I J O P M N
D|D A B C H E F G L I J K P M N O D|D A B C H E F G L I J K P M N O
E|E F G H A B C D M N O P I J K L E|E F G H A B C D M N O P I J K L
F|F G H E B C D A N O P M J K L I F|F G H E B C D A N O P M J K L I
G|G H E F C D A B O P M N K L I J G|G H E F C D A B O P M N K L I J
H|H E F G D A B C P M N O L I J K H|H E F G D A B C P M N O L I J K
I|I N K P M J O L A F C H E B G D I|I L K J M P O N C B A D G F E H
J|J O L M N K P I B G D E F C H A J|J I L K N M P O D C B A H G F E
K|K P I N O L M J C H A F G D E B K|K J I L O N M P A D C B E H G F
L|L M J O P I N K D E B G H A F C L|L K J I P O N M B A D C F E H G
M|M J O L I N K P E B G D A F C H M|M P O N I L K J G F E H C B A D
N|N K P I J O L M F C H A B G D E N|N M P O J I L K H G F E D C B A
O|O L M J K P I N G D E B C H A F O|O N M P K J I L E H G F A D C B
P|P I N K L M J O H A F C D E B G P|P O N M L K J I F E H G B A D C
_________________________________ _________________________________
A|A B C D E F G H I J K L M N O P A|A B C D E F G H I J K L M N O P
B|B C D A F G H E J K L I N O P M B|B C D A F G H E J K L I N O P M
C|C D A B G H E F K L I J O P M N C|C D A B G H E F K L I J O P M N
D|D A B C H E F G L I J K P M N O D|D A B C H E F G L I J K P M N O
E|E F G H A B C D M N O P I J K L E|E F G H A B C D M N O P I J K L
F|F G H E B C D A N O P M J K L I F|F G H E B C D A N O P M J K L I
G|G H E F C D A B O P M N K L I J G|G H E F C D A B O P M N K L I J
H|H E F G D A B C P M N O L I J K H|H E F G D A B C P M N O L I J K
I|I N K P M J O L C H A F G D E B I|I J K L M N O P E F G H A B C D
J|J O L M N K P I D E B G H A F C J|J K L I N O P M F G H E B C D A
K|K P I N O L M J A F C H E B G D K|K L I J O P M N G H E F C D A B
L|L M J O P I N K B G D E F C H A L|L I J K P M N O H E F G D A B C
M|M J O L I N K P G D E B C H A F M|M N O P I J K L A B C D E F G H
N|N K P I J O L M H A F C D E B G N|N O P M J K L I B C D A F G H E
O|O L M J K P I N E B G D A F C H O|O P M N K L I J C D A B G H E F
P|P I N K L M J O F C H A B G D E P|P M N O L I J K D A B C H E F G
_________________________________ _________________________________
A|A B C D E F G H I J K L M N O P A|A B C D E F G H I J K L M N O P
B|B C D A F G H E J K L I N O P M B|B C D A F G H E J K L I N O P M
C|C D A B G H E F K L I J O P M N C|C D A B G H E F K L I J O P M N
D|D A B C H E F G L I J K P M N O D|D A B C H E F G L I J K P M N O
E|E F G H A B C D M N O P I J K L E|E H G F A D C B M P O N I L K J
F|F G H E B C D A N O P M J K L I F|F E H G B A D C N M P O J I L K
G|G H E F C D A B O P M N K L I J G|G F E H C B A D O N M P K J I L
H|H E F G D A B C P M N O L I J K H|H G F E D C B A P O N M L K J I
I|I L K J M P O N E H G F A D C B I|I J K L O P M N A B C D G H E F
J|J I L K N M P O F E H G B A D C J|J K L I P M N O B C D A H E F G
K|K J I L O N M P G F E H C B A D K|K L I J M N O P C D A B E F G H
L|L K J I P O N M H G F E D C B A L|L I J K N O P M D A B C F G H E
M|M P O N I L K J A D C B E H G F M|M P O N K J I L E H G F C B A D
N|N M P O J I L K B A D C F E H G N|N M P O L K J I F E H G D C B A
O|O N M P K J I L C B A D G F E H O|O N M P I L K J G F E H A D C B
P|P O N M L K J I D C B A H G F E P|P O N M J I L K H G F E B A D C
_________________________________ _________________________________
A|A B C D E F G H I J K L M N O P A|A B C D E F G H I J K L M N O P
B|B C D A F G H E J K L I N O P M B|B C D A F G H E J K L I N O P M
C|C D A B G H E F K L I J O P M N C|C D A B G H E F K L I J O P M N
D|D A B C H E F G L I J K P M N O D|D A B C H E F G L I J K P M N O
E|E H G F A D C B M P O N I L K J E|E I G K A M C O B N D P F J H L
F|F E H G B A D C N M P O J I L K F|F J H L B N D P C O A M G K E I
G|G F E H C B A D O N M P K J I L G|G K E I C O A M D P B N H L F J
H|H G F E D C B A P O N M L K J I H|H L F J D P B N A M C O E I G K
I|I L K J M P O N A D C B E H G F I|I G K E M C O A N D P B J H L F
J|J I L K N M P O B A D C F E H G J|J H L F N D P B O A M C K E I G
K|K J I L O N M P C B A D G F E H K|K E I G O A M C P B N D L F J H
L|L K J I P O N M D C B A H G F E L|L F J H P B N D M C O A I G K E
M|M N O P I J K L E F G H A B C D M|M N O P I J K L G H E F C D A B
N|N O P M J K L I F G H E B C D A N|N O P M J K L I H E F G D A B C
O|O P M N K L I J G H E F C D A B O|O P M N K L I J E F G H A B C D
P|P M N O L I J K H E F G D A B C P|P M N O L I J K F G H E B C D A
_________________________________ _________________________________
A|A B C D E F G H I J K L M N O P A|A B C D E F G H I J K L M N O P
B|B C D A F G H E J K L I N O P M B|B C D E F G H A J K L M N O P I
C|C D A B G H E F K L I J O P M N C|C D E F G H A B K L M N O P I J
D|D A B C H E F G L I J K P M N O D|D E F G H A B C L M N O P I J K
E|E H G F C B A D M P O N K J I L E|E F G H A B C D M N O P I J K L
F|F E H G D C B A N M P O L K J I F|F G H A B C D E N O P I J K L M
G|G F E H A D C B O N M P I L K J G|G H A B C D E F O P I J K L M N
H|H G F E B A D C P O N M J I L K H|H A B C D E F G P I J K L M N O
I|I J K L O P M N A B C D G H E F I|I J K L M N O P A B C D E F G H
J|J K L I P M N O B C D A H E F G J|J K L M N O P I B C D E F G H A
K|K L I J M N O P C D A B E F G H K|K L M N O P I J C D E F G H A B
L|L I J K N O P M D A B C F G H E L|L M N O P I J K D E F G H A B C
M|M P O N I L K J E H G F A D C B M|M N O P I J K L E F G H A B C D
N|N M P O J I L K F E H G B A D C N|N O P I J K L M F G H A B C D E
O|O N M P K J I L G F E H C B A D O|O P I J K L M N G H A B C D E F
P|P O N M L K J I H G F E D C B A P|P I J K L M N O H A B C D E F G
_________________________________ _________________________________
A|A B C D E F G H I J K L M N O P A|A B C D E F G H I J K L M N O P
B|B C D E F G H A J K L M N O P I B|B C D E F G H A J K L M N O P I
C|C D E F G H A B K L M N O P I J C|C D E F G H A B K L M N O P I J
D|D E F G H A B C L M N O P I J K D|D E F G H A B C L M N O P I J K
E|E F G H A B C D M N O P I J K L E|E F G H A B C D M N O P I J K L
F|F G H A B C D E N O P I J K L M F|F G H A B C D E N O P I J K L M
G|G H A B C D E F O P I J K L M N G|G H A B C D E F O P I J K L M N
H|H A B C D E F G P I J K L M N O H|H A B C D E F G P I J K L M N O
I|I L O J M P K N A D G B E H C F I|I N K P M J O L A F C H E B G D
J|J M P K N I L O B E H C F A D G J|J O L I N K P M B G D A F C H E
K|K N I L O J M P C F A D G B E H K|K P M J O L I N C H E B G D A F
L|L O J M P K N I D G B E H C F A L|L I N K P M J O D A F C H E B G
M|M P K N I L O J E H C F A D G B M|M J O L I N K P E B G D A F C H
N|N I L O J M P K F A D G B E H C N|N K P M J O L I F C H E B G D A
O|O J M P K N I L G B E H C F A D O|O L I N K P M J G D A F C H E B
P|P K N I L O J M H C F A D G B E P|P M J O L I N K H E B G D A F C
_________________________________ _________________________________
A|A B C D E F G H I J K L M N O P A|A B C D E F G H I J K L M N O P
B|B C D E F G H A J K L M N O P I B|B C D E F G H A J K L M N O P I
C|C D E F G H A B K L M N O P I J C|C D E F G H A B K L M N O P I J
D|D E F G H A B C L M N O P I J K D|D E F G H A B C L M N O P I J K
E|E F G H A B C D M N O P I J K L E|E F G H A B C D M N O P I J K L
F|F G H A B C D E N O P I J K L M F|F G H A B C D E N O P I J K L M
G|G H A B C D E F O P I J K L M N G|G H A B C D E F O P I J K L M N
H|H A B C D E F G P I J K L M N O H|H A B C D E F G P I J K L M N O
I|I P O N M L K J A H G F E D C B I|I P O N M L K J E D C B A H G F
J|J I P O N M L K B A H G F E D C J|J I P O N M L K F E D C B A H G
K|K J I P O N M L C B A H G F E D K|K J I P O N M L G F E D C B A H
L|L K J I P O N M D C B A H G F E L|L K J I P O N M H G F E D C B A
M|M L K J I P O N E D C B A H G F M|M L K J I P O N A H G F E D C B
N|N M L K J I P O F E D C B A H G N|N M L K J I P O B A H G F E D C
O|O N M L K J I P G F E D C B A H O|O N M L K J I P C B A H G F E D
P|P O N M L K J I H G F E D C B A P|P O N M L K J I D C B A H G F E
_________________________________
|A B C D E F G H I J K L M N O P
A|A B C D E F G H I J K L M N O P
B|B C D E F G H I J K L M N O P A
C|C D E F G H I J K L M N O P A B
D|D E F G H I J K L M N O P A B C
E|E F G H I J K L M N O P A B C D
F|F G H I J K L M N O P A B C D E
G|G H I J K L M N O P A B C D E F
H|H I J K L M N O P A B C D E F G
I|I J K L M N O P A B C D E F G H
J|J K L M N O P A B C D E F G H I
K|K L M N O P A B C D E F G H I J
L|L M N O P A B C D E F G H I J K
M|M N O P A B C D E F G H I J K L
N|N O P A B C D E F G H I J K L M
O|O P A B C D E F G H I J K L M N
P|P A B C D E F G H I J K L M N O
-- Kenneth Almquist <ka@cbnewsh.ATT.COM>
Sorry, but Rotman's book on group theory proves that there are
only 5 groups of order 12 and 14 groups of order 16 (not 6 and 19
respectively). Some of the groups generated here are surely
isomorphic. Can one derive all the groups of order 16 by
elementary means? I tried this with a class some 9 years
ago and we got 13 of them, but ran into an intractable
existence problem. I know this can be done easily if
one has the theory of extensions at one's fingertips.
-- Ray Steiner <steiner@bgsuvax.UUCP>
You're right, there was a bug in my routine for testing isomorphism.
The fifth group of order 12 and the ninth through thirteenth groups of
order 16 (reading left to right) are isomorphic to earlier groups.
It's hard to test a program when you don't know what the output is
supposed to be.
-- Kenneth Almquist <ka@cbnewsh.ATT.COM>