Basic Examples 
(5)
 

Find the multiplicative group of integers modulo 5:
In[22]:=
[◼]
MultiplicativeGroup
[5]
Out[22]=
CyclicGroup[4]
See its group multiplication table:
In[23]:=
GroupMultiplicationTable[%]//MatrixForm
Out[23]//MatrixForm=
1
2
3
4
2
3
4
1
3
4
1
2
4
1
2
3
_________________________________________________________________________________________________________________
See the elements of
U
10
:
In[65]:=
GroupElements@
[◼]
MultiplicativeGroup
[10]
Out[65]=
{Cycles[{}],Cycles[{{1,2,3,4}}],Cycles[{{1,3},{2,4}}],Cycles[{{1,4,3,2}}]}
_________________________________________________________________________________________________________________
Compute the multiplicative group of various integers:
In[24]:=
[◼]
MultiplicativeGroup
/@Range[10]
Out[24]=
{CyclicGroup[1],CyclicGroup[1],CyclicGroup[2],CyclicGroup[2],CyclicGroup[4],CyclicGroup[2],CyclicGroup[6],AbelianGroup[{2,2}],CyclicGroup[6],CyclicGroup[4]}
_________________________________________________________________________________________________________________
Multiple cyclic factors emerge for composite inputs:
In[25]:=
[◼]
MultiplicativeGroup
[12]
Out[25]=
AbelianGroup[{2,2}]
_________________________________________________________________________________________________________________
Calculate large groups:
In[71]:=
AbsoluteTiming@
[◼]
MultiplicativeGroup
[50!]
Out[71]=
{0.0009268,AbelianGroup[{2,28,30,36,40,42,46,272,342,506,2028,13310,4941258,195312500,20920706406,35184372088832}]}
Calculate its order:
In[72]:=
GroupOrder[%[[2]]]
Out[72]=
4218559200885839042679312107816703841788854953574400000000000000

Properties and Relations 
(7)
 


Neat Examples 
(1)
 
