I’m reading a text on group theory from the 1960’s, and I see this:
G = <a> has order n. a^k is a generator of G iff (k,n) = 1.
What does (k,n) mean? Is this another way of writing the binomial coefficient (nCk)? It doesn’t seem to be defined anywhere in the book, and an Internet search isn’t helpful.
(k, n) is oftentimes used as shorthand for “greatest common divisor of k and n,” or at least it was in the math courses I’ve recently taken. (So, it’s not necessarily archaic.) In this case, “(k,n) = 1” just means k and n are relatively prime.
Ah, thanks. I’ve only ever seen gcd(k,n) used.
Which book are you looking at? I’ve seen both gcd(k, n) and (k, n) used. I’ve also seen [k, n] to represent the least common multiple of k and n, in case you run into that.
An Introduction to the Theory of Groups by Rotman. Actually, looking at it, it’s the third edition, so it was published in the early 80’s, whilst the first was published in the 60’s.
Huh. I just checked my copy of Herstein (a more mainstream introductory algebra text), and he uses the (a, b) notation as well.
If I remember correctly, Hungerford’s Abstract Algebra: An Introduction, written in 1996, uses that notation for the greatest common denominator as well.
Another comment about the notation:
Given a ring R and a collection of elements x_1, x_2,…, x_n, it’s fairly common to denote the ideal generated by the set {x_1, x_2,…, x_n} as (x_1, x_2,…, x_n).
The integers are a principal ideal domain, so every ideal is generated by a single element. If {x_1, x_2,…, x_n} is a subset of the integers, then the ideal generated by that set, (x_1, x_2,…, x_n), is the same as the principal ideal (d), where d is the gcd of x_1, x_2,…, x_n.
So the (a,b) notation for greatest common divisor coincides with the (a,b) notation for the ideal generated by a and b.