# Terminology: center (of a group, of a ring, …)

What is the etymology of the word "center" as used in abstract algebra, e.g. the center of a group, or of an algebra?

My best guess is that it might've come from matrix algebras, where often the center just consists of (scalar) diagonal matrices, whose nonzero coefficients appear in the "center" of the matrix.

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May I be so bold and add the question where the word "trace" comes from? As trade-off, I might note that the word "ring" in it's german origin is not referring to a round ring, like one you wear on your finger, but it has also the meaning of a union, so it's similar to a group. For example "Verbrecherring" is a community or circle of organized crime. –  NikolajK Feb 13 '12 at 14:06
I have shared this question at G+. Let's hope we get a good answer :) –  user21436 Feb 13 '12 at 14:43
For "trace", see jeff560.tripod.com/t.html –  Robert Israel Feb 13 '12 at 14:46
Also for "ring", jeff560.tripod.com/r.html –  Robert Israel Feb 13 '12 at 14:47
Geez,where's John Stillwell when we need him-this question is really in his wheelhouse.......... –  Mathemagician1234 Feb 13 '12 at 16:45

## 2 Answers

I will give this a go:

This term could have been borrowed from Latin of the late $14^{th}$ century:

L. centrum "center," originally fixed point of the two points of a compass,

Since the elements of the center of a group fix all elements under conjugation, this might have been a motivation.

Might be of interest:

from Greek. kentron "sharp point, goad, sting of a wasp," from kentein "stitch," from PIE base *kent- "to prick" (cf. Breton kentr "a spur," Welsh cethr "nail," O.H.G. hantag "sharp, pointed"). The verb is from 1590s. Spelling with -re popularized in Britain by Johnson's dictionary, though -er is older. Related: Centered; centering. Center of gravity is recorded from 1650s.

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From what I can tell, the word 'center' (or its German friend 'zentrum,' which is what gave us the notation $Z(G)$ that we use today) was introduced by JA de Seguier, in his 1904 textbook Elements de la theorie des groups abstraits.

This is interesting because it was also this textbook that thought to have created the name 'demi-groupe,' which later became semi-group, and eventually what we now call a semigroup.

I don't know of the original rationale behind choosing the word center, but there are a few possibilities that strike me. The ideas of algebras, algebra homomorphisms, and automorphisms were known. Here's a thought: were there yet 'inner automorphisms?' The phrase 'inner product' had been around since 1844, when Grassman developed different algebras of the hypercomplex numbers (in Die lineale Ausdehnungslehre, 1844). Interestingly, it seems that he chose these names more for their being antonyms, rather than an intrinsic meaning.

It happens to be that those simple algebras over a field with the property that every automorphism is an inner automorphism are precisely those that are called central (where the center as we know it today is exactly the base field). I like the idea that central and inner are related (even though it's not entirely founded).

In a similar vein, actions were in high fashion and few actions are as well-studied as conjugation. Kannappan's suggestion that those elements fixing elements under conjugation are the source of the name center seems reasonable as well.

But it is also possible that the naming convention falls closer to Polya than Grassman. Polya coined the term 'central limit theorem' in his paper (Über den zentralen Grenzwertsatz der Wahrscheinlichkeitsrechnung und das Momentenproblem), but it's believed that he named it 'central' because of its 'central importance' rather than any intrinsic quality. (Though it fits nicely with the intuition of things called measures of central tendency).

But these rationales are speculative.

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Is it possible that the name relates to the concept of orbit? –  Asaf Karagila Apr 13 '12 at 22:54