301.3E Centralizer of an Element of a Group

The centralizer of an element a in a group G is the set of all elements of G that commute with a. Definition, example, and how to keep abelian, center, and centralizer definitions straight.

[@CraaaabPeople]

Well done! Made the concepts of center and centralizer very intuitive.

[@horaciormartinez1551]

Great !!! Thank you !!!

[@PunmasterSTP]

Centralizer? More like "Cool video; now we're wiser!"

[@julimate]

Your explanation was great but I wasn't able to find the link to the dihedral group explorer you used in this video :c

[@192ali1]

Greetings. As always, thank you for excellent video lectures. Question?: 9:50-9:55
If centralizers are subgroups, as you said they are, then the smallest non trivial centralizer has two elements, e and the second element, provided the second elements is its own inverse. If the element beside e is not its own inverse, then the smallest centralizer should have three elements. Am I correct?

[@MatthewSalomone]

Ali Umar Yes. Since the centralizer of g in G is a subgroup of G, it may have the structure of any known group. So what you say here is a more generally true statement about groups: any group having an element h that is not its own inverse must have at least three elements, namely e, h, and h⁻¹. (By the way, groups in which *every* element is its own inverse are called elementary groups. They're all abelian and have order equal to a power of 2.)

[@192ali1]

@@MatthewSalomone Thank you Sir. I love each and every single of your lectures. You are an assets to your institution and to your students. They are fortunate to have your live lectures and I am fortunate to learn about your you-tube channel and to take notes of your excellent video lectures. Stay Safe

[@NeillClift]

When you say the centralizer is the largest set of elements that commute I start thinking there are multiple sets and you pick the largest. My thinking is that you are really saying that its the set of all g that commute with a.