U@10D
y
Z@4D
y
*
1
3
7
9
+
0
1
2
3
1
1
3
7
9
0
0
1
2
3
3
3
9
1
7
1
1
2
3
0
7
7
1
9
3
2
2
3
0
1
9
9
7
3
1
3
3
0
1
2
Out[8]=
x
x
"U@10D"
y
x
*
1
3
9
7
1
1
3
9
7
3
3
9
7
1
9
9
7
1
3
7
7
1
3
9
U@8D
y
*
1
3
5
7
1
1
3
5
7
3
3
1
7
5
5
5
7
1
3
7
7
5
3
1
Out[17]=
x
Group properties Element properties Group Calculator
Klein4
y
*
e
a
b
c
e
e
a
b
c
a
a
e
c
b
b
b
c
e
a
c
c
b
a
e
Out[19]=
x
2
Lab2Student.nb
In[20]:=
G6 = Z@6D
G7 = U@7D
CayleyTable@8G6, G7<, Mode ® VisualD
Out[20]=
Groupoid@80, 1, 2, 3, 4, 5<, Mod@ð1 + ð2, 6D &D
Out[21]=
Groupoid@81, 2, 3, 4, 5, 6<, Mod@ð1 ð2, 7D &D
Group properties Element properties Group Calculator
Group properties Element properties Group Calculator
Lab2Student.nb
"Z@6D"
y
"U@7D"
y
+
0
1
2
3
4
5
*
1
2
3
4
5
6
0
0
1
2
3
4
5
1
1
2
3
4
5
6
1
1
2
3
4
5
0
2
2
4
6
1
3
5
2
2
3
4
5
0
1
3
3
6
2
5
1
4
3
3
4
5
0
1
2
4
4
1
5
2
6
3
4
4
5
0
1
2
3
5
5
3
1
6
4
2
5
5
0
1
2
3
4
6
6
5
4
3
2
1
In[27]:=
x
x
CayleyTable@G7, TheSet ® 81, 5, 2, 6, 4, 3<, Mode ® VisualD
Z@6D
y
U@7D
y
+
0
1
2
3
4
5
*
1
2
3
4
5
6
0
0
1
2
3
4
5
1
1
2
3
4
5
6
1
1
2
3
4
5
0
2
2
4
6
1
3
5
2
2
3
4
5
0
1
3
3
6
2
5
1
4
3
3
4
5
0
1
2
4
4
1
5
2
6
3
4
4
5
0
1
2
3
5
5
3
1
6
4
2
5
5
0
1
2
3
4
6
6
5
4
3
2
1
Out[27]=
x
x
Group properties Element properties Group Calculator
U@7D
y
*
1
5
2
6
4
3
1
1
5
2
6
4
3
5
5
4
3
2
6
1
2
2
3
4
5
1
6
6
6
2
5
1
3
4
4
4
6
1
3
2
5
3
3
1
6
4
5
2
Out[28]=
x
In[29]:=
CayleyTable@G7, TheSet ® 81, 3, 2, 6, 4, 5<, Mode ® VisualD
Group properties Element properties Group Calculator
3
4
Lab2Student.nb
U@7D
y
*
1
3
2
6
4
5
1
1
3
2
6
4
5
3
3
2
6
4
5
1
2
2
6
4
5
1
3
6
6
4
5
1
3
2
4
4
5
1
3
2
6
5
5
1
3
2
6
4
Out[29]=
x
In[30]:=
CayleyTable@G7, TheSet ® 81, 3, 4, 6, 2, 5<, Mode ® VisualD
Group properties Element properties Group Calculator
U@7D
y
*
1
3
4
6
2
5
1
1
3
4
6
2
5
3
3
2
5
4
6
1
4
4
5
2
3
1
6
6
6
4
3
1
5
2
2
2
6
1
5
4
3
5
5
1
6
2
3
4
Out[30]=
x
In[31]:=
CayleyTable@G7, TheSet ® 81, 5, 4, 6, 2, 3<, Mode ® VisualD
Group properties Element properties Group Calculator
U@7D
y
*
1
5
4
6
2
3
1
1
5
4
6
2
3
5
5
4
6
2
3
1
4
4
6
2
3
1
5
6
6
2
3
1
5
4
2
2
3
1
5
4
6
3
3
1
5
4
6
2
Out[31]=
x