Elements of semigroup commuting with subgroup

  • Context: Graduate 
  • Thread starter Thread starter mnb96
  • Start date Start date
  • Tags Tags
    Elements Subgroup
Click For Summary

Discussion Overview

The discussion revolves around the properties of a semigroup S that contains a subgroup G, specifically focusing on an element s in S that commutes with all elements of G. Participants explore the implications of this relationship and whether it leads to further conclusions about the nature of s within the context of abstract algebra.

Discussion Character

  • Exploratory, Technical explanation, Conceptual clarification

Main Points Raised

  • One participant questions whether the property of an element s commuting with all elements of a subgroup G implies any further statements beyond its commutation.
  • Another participant suggests that s can be classified as being in the centralizer of G in S.
  • A further contribution notes that the concept of centralizer can be applied to subsets in general, not limited to subgroups.
  • One participant expresses appreciation for the clarification regarding the centralizer.
  • A later post introduces a tangential idea about the distinction between terminology in science fiction and abstract algebra, suggesting a potential insight on the topic.

Areas of Agreement / Disagreement

Participants appear to agree on the definition of the centralizer in relation to the subgroup G, but the initial question about the implications of the commutation property remains open and unresolved.

Contextual Notes

The discussion does not resolve whether the commutation of s with G leads to any additional conclusions about s, nor does it clarify the implications of the centralizer concept in broader contexts.

Who May Find This Useful

Readers interested in abstract algebra, particularly those exploring semigroups and subgroup properties, may find this discussion relevant.

mnb96
Messages
711
Reaction score
5
Hello,

Suppose we have a semigroup S with a subgroup G≤S.
Assume there is an element s∈S that commutes with all the elements in G. Does this statement implies (or is equivalent to) another statement?

If hypothetically the element s would have been in G, then we could have said that s was an element of the center of G, but since s is not contained in G then I suspect we can't say much more than just saying that it commutes with G, or can we?
 
Physics news on Phys.org
You can say that s is in the centralizer of G in S.
 
And the term extends to subsets in general, not just to subgroups.
 
Thanks a lot for your answers, especially for pointing out the centralizer.
 
mnb96 said:
Thanks a lot for your answers, especially for pointing out the centralizer.
I am thinking of starting an " insight" pointing out the difference between science fiction terms and abstract algebra ones. Phaser: Algebra or Sci-Fi?
 

Similar threads

Undergrad Meaning of "passage to the quotient"?
  • · Replies 3 ·
Replies
3
Views
1K
Undergrad Question about "A Group Epimorphism is Surjective"
  • · Replies 13 ·
Replies
13
Views
2K
Undergrad Differences between left vs right actions in some group theory questions
  • · Replies 1 ·
Replies
1
Views
1K
Undergrad Trouble with a passage in Zariski & Samuel's Commutative algebra text
  • · Replies 5 ·
Replies
5
Views
2K
Undergrad Equivalent definitons for primitive polynomials
  • · Replies 24 ·
Replies
24
Views
2K
Graduate What is a Commutator Subgroup?
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Dfns group action & group, written in terms of the other
  • · Replies 26 ·
Replies
26
Views
1K
Graduate Normalizers and normal subgroups.
  • · Replies 3 ·
Replies
3
Views
3K
Undergrad Meaning of "identify" & variations in different math context
  • · Replies 5 ·
Replies
5
Views
1K
MHB Is \( f(P) \) a \( p \)-Sylow Subgroup of \( H \)?
  • · Replies 3 ·
Replies
3
Views
2K