As mentioned in a previous question of mine, I am working through Introduction to the Theory of Groups, by Joseph Rotman (fourth edition). This question is in a similar vein as the previous. On the section on the Jordan-Holder Theorem (pg. 98), the book defines a refinement of a normal series as follows
Given a group $G$ and a normal series:
$$G=H_0 \geq H_1 \geq H_2 … \geq H_m=1$$
A refinement of said normal series is:
$$G=G_0 \geq G_1 \geq G_2 … \geq G_n=1$$
if $G_0, G_1, … , G_n$ is a subsequence of $H_0, H_1, … , H_m$.
According to this, the refinement should be shorter or equal in length compared to the original normal series, but on the next page, where it proves the Schreier Refinement Theorem, it gives an example of a refinement with a longer length than the original. Is this another typo? If so, is there a known collection of all the errors for this book? If not, what am I missing?
Best Answer
There is no typo. The problem here is that you have misread/misstated what the definition actually says.
Here is the statement in my copy of the book (same edition, 9th printing):
Read carefully: the $H_i$ form a refinement of the sequence given by the $G_j$. You misquoted it/misunderstood it as saying that the $G_j$ were a refinement of the sequence given by the $H_i$. That is not what it says.
So it makes sense that the $G_i$ are a subsequence of the $H_j$: there are more $H_j$ than $G_i$.
As to errata: my copy did indeed come with a piece of paper labeled "Errata for Rotman, An Introduction to the Theory of Groups, Fourth Edition. The first erratum listed is on page 43. I can't find it on the publisher's website (which is where I would expect a link to that) and of course the author is sadly no longer with us.