#### August 1989 Posting to USENET newsgroup SCI.MATH by Kenneth Almquist

>From sdcsvax!ucsdhub!hp-sdd!ncr-sd!ncrcae!hubcap!gatech!purdue!tut.cis.ohio-
state.edu!att!cbnewsh!ka@cbnewsh.ATT.COM Mon Aug 28 09:45:30 PDT 1989
Article 8170 of sci.math:
Path: sdcsvax!ucsdhub!hp-sdd!ncr-sd!ncrcae!hubcap!gatech!purdue!tut.cis.ohio-
state.edu!att!cbnewsh!ka@cbnewsh.ATT.COM
>From: ka@cbnewsh.ATT.COM (Kenneth Almquist)
Newsgroups: sci.math
Subject: All groups of orders 1 through 16
Message-ID: <3368@cbnewsh.ATT.COM>
Date: 26 Aug 89 22:10:01 GMT
Sender: jgy@cbnewsh.ATT.COM
Lines: 437

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.

Kenneth Almquist

``` ---------------------------------
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-communtative      1   2    1     4     1    14```
```   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
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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
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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
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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
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