"We were learning about centralizers in my abstract..." was answered by our experts

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Algebra
Abstract algebra

Essence Ingram

Answered question

2022-04-16

We were learning about centralizers in my abstract algebra class, and I had a thought about centralizers.
Is every subgroup of a group G the centralizer of some element? And if not, how many of the subgroups are?

Answer & Explanation

glanzerjbdo

Beginner2022-04-17Added 13 answers

If G is an abelian group and if

a \in G

, then the centralizer of a is G itself. Therefore, if H is a subgroup of G which is not G itself, then H cannot be the centralizer of an element of G.

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