Elements of semigroup commuting with subgroup

In summary, the conversation discusses the relationship between a semigroup and its subgroup, and the concept of an element that commutes with all elements in the subgroup. It is mentioned that this element could be considered part of the center of the subgroup, but since it is not contained in the subgroup, it is instead considered to be in the centralizer of the subgroup. The conversation also touches on the use of science fiction terms in abstract algebra.
  • #1
mnb96
715
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?
 
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  • #2
You can say that s is in the centralizer of G in S.
 
  • #3
And the term extends to subsets in general, not just to subgroups.
 
  • #4
Thanks a lot for your answers, especially for pointing out the centralizer.
 
  • #5
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?
 

What is a semigroup?

A semigroup is a mathematical structure that consists of a set of elements and an operation between any two elements in the set that is associative. This means that the order in which the operations are performed does not matter.

What is a subgroup?

A subgroup is a subset of a semigroup that forms a semigroup itself, meaning that it also follows the associativity property. In other words, the elements in a subgroup also commute with each other.

What does it mean for an element to commute with a subgroup?

An element in a semigroup commutes with a subgroup if the element and every element in the subgroup can be interchanged without changing the result of the operation. This is a stronger condition than just being in the subgroup, as it requires all elements in the subgroup to commute with the element.

Why is it important to study elements of semigroups that commute with subgroups?

Studying elements of semigroups that commute with subgroups can provide insights into the structure and properties of the semigroup. It can also help in solving problems and making connections with other areas of mathematics, such as group theory and algebraic structures.

What are some examples of elements of semigroups that commute with subgroups?

Some examples include the identity element, which commutes with any subgroup, and inverses of elements in the subgroup, which also commute with the subgroup. Other examples can be found by studying specific semigroups and their subgroups, such as the subgroups of a free semigroup.

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