Basic Examples (5)
Basic Examples
(5)
Embed [2] in [4] via :
(12)↦(13)(24)
In[22]:=
ϕ=[CyclicGroup[2],CyclicGroup[4],{Cycles[{{1,3},{2,4}}]}]
Out[22]=
Click on + to reveal more information about the homomorphism. For example, is injective but not surjective. Obtain the underlying mapping:
ϕ
In[23]:=
Normal[ϕ]
Out[23]=
Cycles[{}]Cycles[{}],Cycles[{{1,2}}]Cycles[{{1,3},{2,4}}]
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Retrieve the domain or co-domain of a homomorphism:
In[24]:=
ϕ["Domain"]
Out[24]=
CyclicGroup[2]
In[25]:=
ϕ["CoDomain"]
Out[25]=
CyclicGroup[4]
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Find the kernel of this homomorphism, indeed the kernel is singleton thus is an embedding:
ϕ
In[26]:=
ϕ["Kernel"]
Out[26]=
PermutationGroup[{}]
This is the trivial group:
In[27]:=
GroupElements[%]
Out[27]=
{Cycles[{}]}
Find the image group:
In[28]:=
ϕ["Image"]
Out[28]=
PermutationGroup[{Cycles[{{1,3},{2,4}}]}]
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Try to specify an invalid homomorphism:
In[29]:=
Out[29]=
$Failed
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The package efficiently works with large groups. For example, here is the identity homomorphism from [10] to itself:
In[30]:=
id=AbsoluteTiming@[SymmetricGroup[20],SymmetricGroup[20],GroupGenerators[SymmetricGroup[20]]]
Out[30]=
0.0028744,
Its image and kernel groups:
In[31]:=
AbsoluteTiming@id[[2]]["Image"]
Out[31]=
{0.0000263,PermutationGroup[{Cycles[{{1,2}}],Cycles[{{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20}}]}]}
In[32]:=
AbsoluteTiming@id[[2]]["Kernel"]
Out[32]=
{0.217765,PermutationGroup[{}]}
Scope (2)
Scope
(2)
Applications (3)
Applications
(3)
Properties and Relations (4)
Properties and Relations
(4)
Possible Issues (3)
Possible Issues
(3)