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GroupCentralizer
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GroupCentralizer

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GroupCentralizer[group,g]

returns the centralizer of the element g in group.

Details

  • The centralizer of an element g in group is the subgroup of elements of group that commute with g.

Examples

open allclose all

Basic Examples  (1)Summary of the most common use cases

Take a permutation group:

In[1]:=1
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https://wolfram.com/xid/01y9pyqc39j-6hym04
Out[1]=1

And a permutation:

In[2]:=2
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https://wolfram.com/xid/01y9pyqc39j-5pghl1

This is the subgroup of permutations in the group that commutes with the permutation g:

In[3]:=3
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https://wolfram.com/xid/01y9pyqc39j-h4ye6x
Out[3]=3
In[4]:=4
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https://wolfram.com/xid/01y9pyqc39j-w6dfrk
Out[4]=4

Scope  (1)Survey of the scope of standard use cases

Take a permutation group:

In[1]:=1
✖
https://wolfram.com/xid/01y9pyqc39j-w5mqi
Out[1]=1

And a permutation:

In[2]:=2
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https://wolfram.com/xid/01y9pyqc39j-4hr5wq

This is the subgroup of permutations in the group that commutes with the permutation g:

In[3]:=3
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https://wolfram.com/xid/01y9pyqc39j-k0b4c9
Out[3]=3
In[4]:=4
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https://wolfram.com/xid/01y9pyqc39j-rto6ob
Out[4]=4

Check the result by direct computation of products with all elements:

In[5]:=5
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https://wolfram.com/xid/01y9pyqc39j-4q05k1
Out[5]=5

This function checks that a permutation h commutes with g:

In[6]:=6
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https://wolfram.com/xid/01y9pyqc39j-1bpyva

All 16 permutations in the centralizer commute with g:

In[7]:=7
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https://wolfram.com/xid/01y9pyqc39j-dsdh7u
Out[7]=7
Wolfram Research (2010), GroupCentralizer, Wolfram Language function, https://reference.wolfram.com/language/ref/GroupCentralizer.html.
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Wolfram Research (2010), GroupCentralizer, Wolfram Language function, https://reference.wolfram.com/language/ref/GroupCentralizer.html.

Text

Wolfram Research (2010), GroupCentralizer, Wolfram Language function, https://reference.wolfram.com/language/ref/GroupCentralizer.html.

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Wolfram Research (2010), GroupCentralizer, Wolfram Language function, https://reference.wolfram.com/language/ref/GroupCentralizer.html.

CMS

Wolfram Language. 2010. "GroupCentralizer." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/GroupCentralizer.html.

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Wolfram Language. 2010. "GroupCentralizer." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/GroupCentralizer.html.

APA

Wolfram Language. (2010). GroupCentralizer. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/GroupCentralizer.html

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Wolfram Language. (2010). GroupCentralizer. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/GroupCentralizer.html

BibTeX

@misc{reference.wolfram_2025_groupcentralizer, author="Wolfram Research", title="{GroupCentralizer}", year="2010", howpublished="\url{https://reference.wolfram.com/language/ref/GroupCentralizer.html}", note=[Accessed: 29-March-2025 ]}

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@misc{reference.wolfram_2025_groupcentralizer, author="Wolfram Research", title="{GroupCentralizer}", year="2010", howpublished="\url{https://reference.wolfram.com/language/ref/GroupCentralizer.html}", note=[Accessed: 29-March-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_groupcentralizer, organization={Wolfram Research}, title={GroupCentralizer}, year={2010}, url={https://reference.wolfram.com/language/ref/GroupCentralizer.html}, note=[Accessed: 29-March-2025 ]}

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@online{reference.wolfram_2025_groupcentralizer, organization={Wolfram Research}, title={GroupCentralizer}, year={2010}, url={https://reference.wolfram.com/language/ref/GroupCentralizer.html}, note=[Accessed: 29-March-2025 ]}