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Elements of semigroup commuting with subgroup

  1. Jan 30, 2016 #1

    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?
  2. jcsd
  3. Jan 30, 2016 #2


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    You can say that s is in the centralizer of G in S.
  4. Jan 30, 2016 #3


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    And the term extends to subsets in general, not just to subgroups.
  5. Jan 30, 2016 #4
    Thanks a lot for your answers, especially for pointing out the centralizer.
  6. Jan 30, 2016 #5


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    I am thinking of starting an " insight" pointing out the difference between science fiction terms and abstract algebra ones. Phaser: Algebra or Sci-Fi?
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