So, I'm studying number theory by myself on a book I found on the internet, I'll leave a link to the pdf, the page i'm refering to is page 96, there is this one lemma stated and proved in the book that I seem to not get at all: "We now prove a lemma that gives us how many incongruent integers can have a given order modulo p, lemma: let p be a prime and m a positive integer such that p-1=m*k for some integer k, then:
S(m)= |{m: 0 < m < p, m ∈ Z}| ≤ φ(m)"
I've studied the order of an integer mod m, but i'm not completely sure what he means by "integer with a given order modulo p", does it count how many a's are there such that a^(ordᵖa)≡1 (mod p)? also, that statement about S(m), could someone try to explain it to me a bit more clearly? i'm really confused...
link of the book's pdf: https://www.saylor.org/site/wp-content/uploads/2013/05/An-Introductory-in-Elementary-Number-Theory.pdf
P.S. I'm in 4th year of highschool I'm studying this stuff because I truly love it, so I might not be familiar with some more advanced stuff, but if I have to be honest this is the first theorem of the book that i find kinda strange...