Usually the word center means the center of a circle. I have encountered the word center in group theory, but do not see any connection with the center of a circle. I think the history of group theory probably has something to do with it.

Can anyone tell me how the terms center and centralizer came up in group theory?

I have searched for "center in group theory" here, but it was not previously asked, and did not get anything from the web either.

  • 4
    $\begingroup$ The English usage of center of a group presumably is merely a translation of the German Zentrum. But that merely moves the question to the reason for the German terminology. I looked in my old German copy of van der Waerden, but he give no reason for the term Zentrum. $\endgroup$ – Gerald Edgar Sep 10 '18 at 11:48
  • 1
    $\begingroup$ We denote the center of a group Z(G) precisely because it came from German practice. $\endgroup$ – Stella Biderman Sep 11 '18 at 12:59
  • 1
    $\begingroup$ Related: math.stackexchange.com/q/3075020/10513 $\endgroup$ – user1729 Jan 25 '19 at 11:14

The center (originally the central) seems to have appeared between the first and second edition of Burnside’s book (1897, §53 vs. 1911, §93) and more precisely in de Séguier (1904, §51):

Ainsi l’ensemble des éléments normaux de $\mathrm G$ est un diviseur normal qui sera dit central de $\mathrm G$.

(Jahrbuch: “Die Gesamtheit invarianter Elemente einer Gruppe nennt Verf. le central der Gruppe.”)

Neither gives an explanation. Guess: conjugacy classes are the orbits of the group’s conjugation action on itself, and a (Kepler) orbit is a point iff it lies at the center. (Then again, maybe they meant the kind of central where operators are busy commuting?)

| improve this answer | |

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.