# Notation: Exponent of a group

The exponent of a group $G$ is the least positive $n$ such that $g^n = e$ for all $g \in G$. This is obviously a sensible name for the concept.

A notational awkwardness arises however when the group $G$ is abelian and written additively. I find it grating to refer to the least positive $n$ such that $\forall g \in G$ $ng = e$ as the exponent because there is nothing going on that even looks like exponentiation.

Is there an alternate terminology that can be used in this situation?

• I guess this is not accepting your frame of question, but my advice is to still say "exponent" (and perhaps commenting on the terminological infelicity). Doing anything else is complicating things needlessly. – Ben Webster Nov 1 '10 at 1:14
• @Ben: The word "period" is standard and is in most algebra books (I gave the example of Lang, only because it is the first book given by google books). Using "exponent" would be weird. – Mark Sapir Nov 1 '10 at 1:29
• I would use "annihilator", since an abelian group is, among other things, a $\mathbb Z$-module. – Theo Johnson-Freyd Nov 1 '10 at 2:21
• Either you're only ever considering abelian groups, in which case Theo's advice seems quite sensible, or you're mostly considering nonabelian groups but occasionally run across an abelian one. In the latter case my advice is: write the group multipliciatively anyway! That is think of Z/4 as g^x for x in Z/4 and g a formal generator. – Noah Snyder Nov 1 '10 at 4:42
• Following Noah's suggestion, you can write $\mathbb{Z}/4\mathbb{Z}$ multiplicatively as $\mu_4(\mathbb{C})$, or formally as $\langle g \mid g^4 = 1\rangle$ – S. Carnahan Jan 6 '11 at 18:18

It is called period if the group is Abelian and the notation is additive, an Abelian group of period $n$: look here (Lang's "Algebra") . The period of an element $a$ (in the additive notation) is sometimes called the order of $a$. See, for example, Fuchs' classic book.

• It seems that for Lang only elements of groups have periods, and these are what most people call orders. – Dylan Moreland Nov 1 '10 at 1:34
• The term "period" is commonly used in a different context; namely, the period of a subset S of an abelian group is the set of all group elements g such that S+g=S. (This is also called the stabilizer of S.) Thus, the period of the whole group is the group itself! I myself use the term "exponent" without experiencing any moral inconvenience :-) – Seva Nov 1 '10 at 11:21
• @Seva: Serial killers don't have any moral problems also. As Dostoevsky used to say: "тварь дрожащая или право имею". Of course you have the right to call anything anyhow. – Mark Sapir Nov 1 '10 at 13:31
• @Mark: you seem to do the common mistake of declaring "standard" what you are used to, and "non-standard" anything else. There are lots of books and papers using the term period the way I described; if you are not familiar with them, this does not mean that they do not exist. And yes, I am very much surprised with your "not quite polite" reaction, and above all, with your advocating of this reaction. – Seva Nov 1 '10 at 19:38
• Please calm down everyone. This question is not important enough to warrant such strong emotions! – Oliver Nov 2 '10 at 20:17

First, I appologise since this should rather be a comment (but I cannot yet comment).

Yet, in the process of reading this question and some of the linked material I got confused, and since I very frequently use the term exponent for abelian groups denoted additively, and know various people who do likewise, I am quite interested in this, too.

1. I looked at the linked content of Fuchs's book but I failed to locate the part that Mark Sapir is referring to. (I have no access to the book at the moment, and thus would be greatful for a precise quote of the relevant part.) The only thing I found online is that 'periodic' is used a synonym for 'torsion', but no mention of period of a group.

2. Also in Lang I was unable to find the mentioned usage, it seems Dylan Moreland made the same experience. Indeed, it seems period of an element is used in Lang also when the notation is multiplicative.

Sorry, for this abuse of 'answer', but I would be grateful for clarification and did not know how to express this except via and 'answer'.