Topics of Group Theory Required to Understand Betti Numbers I am studying Group Theory. I made sure I have a problem at hand, as part of the motivation for my study. I have chosen the problem as being able to understand as well as compute Betti Numbers where the complex is just a simplicial complex. 
Now, I have started preparing a list of topics specifically in Group Theory, that are most importantly required to understand Betti Numbers in Algebraic Topology. I have:
i) Quotient Groups
ii) Rank of a Group (Rank of a Quotient Group, Co-Homology Group)
iii) Chain Group and Basis of Chain Group
iv) Concept of Dual Basis
v) Isomorphic Groups
vi) Concept of Codimension
vii) Direct Sums of Groups (Cohomology Groups)
viii) At the intersection of Group Theory and AT, we have: Homology Groups and Cohomology Groups.
iX) Matrix representation of boundary and co-boundary operators (transpose)
Have I missed anything, or is there a topic you might want to stress on? I am not looking at extreme abstraction, and instead more of looking at a set of abstract group theoretic concepts to be highlighted during my reading in order to have a direction and not be bogged down within the vastness of both the fields. 
Thanks.  
My background: So, this is my background.. I am a statistician(versed in both, mathematical and applied stat.) and have been active on CrossValidated SE.
This was another question I had asked over an year ago here: Rank of a cohomology group, Betti numbers.  This is the story that brought me here to this reading process: "About an year back, I was working on a statistical ranking problem and stumbled on a classical work by Amrtya Sen(Nobel laureate) followed by applied work : hodge decompositions on graphs ("http://arxiv.org/pdf/0811.1067v2.pdf") where they focussed on effects of intransitivities and inconsistencies in rankings and the usage of a discrete version of hodge/helmholtz decomposition to detect these and account for it in least squares (http://arxiv.org/pdf/1011.1716.pdf) and ranking problems which are very famous in statistics"  Following which, given my fascination with the concepts of linear algebra and its wide uses in statistics, I started this journey of reading about algebraic topology out of no where as a basic student, given only my statistcial training and I was surprised that i really started enjoying this reading, which i kept on continuing every now and then and started making notes on my own. 
 A: Are you saying that you have never studied any group theory whatsoever?  (If so, I am somewhat curious about how you managed to construct your solidly plausible list of topics to study.)  
If you really have no prior exposure to group theory -- e.g. the amount of group theory one learns in a first undergraduate algebra course -- then I am concerned that your program of study is a little too specialized.  Here are two special features of homology groups of simplicial complexes:
1) They are always commutative.
2) Assuming the complex is finite, they are finitely generated.
Finitely generated commutative groups are much easier to understand than either non-commutative groups or infinitely generated commutative groups: there is a single structure theorem which tells you everything you need to know in most situations.* Moreover this structure theorem and its proof are closer in spirit to matrix theory than group theory: it is in fact a close cousin of the canonical forms theorems in linear algebra.  
(Further, to reiterate what is mentioned in some of the comments above: most of the topics you list are closer to linear algebra -- or its elder sibling, module theory -- than group theory per se.  A suitably theoretical linear course would prepare you better for basic algebraic topology than any course in group theory I can think of.)
So if you open any text on group theory I can think of, I fear you'll find that most of what you're reading about is manifestly irrelevant to your chosen goal, which could be discouraging.  This makes me think your chosen goal may be a little idiosyncratic/off-center: most of the things one learns in a first group theory course are vital or useful to know later on in one's mathematical life, even if they don't come up in the study of co/homology of simplicial complexes.  If you can understand algebraic topology, then for sure you can learn the basics of group theory in a less mercenary way but still in a reasonable amount of time: say, within a month or two if you're working on it several days a week.  
May I be so bold as to suggest a perturbation of your "problem"?  I suggest that you seek to understand cohomology of groups.  It sits very nicely at the border of algebra and topology, with easy view of other important areas of mathematics.  (As a personal aside, last semester I taught an introductory graduate course on homological algebra.  For the first three months I felt that I was mostly discussing shallow -- but still nontrivial, and at times intricate -- formalities with categories and diagrams.  In the last month I started talking about cohomology of groups, and the whole thing came alive.  In some ways I wish the course had been "cohomology of groups featuring homological algebra" rather than the other way around!)  Kenneth Brown's GTM Cohomology of Groups is a very nice introduction, although it should be read with more basic texts on group theory and topology close at hand.
*: Truly, we live in a golden age of mathematics: nowadays there are brilliant people hard at work on problems involving finite commutative groups, and even finite groups of prime order: the discrete logarithm problem, the Davenport constant, additive combinatorics...But it is a very rare group theory text which will tell you about any of these things.  In fact it is sort of built into the spirit of group theory that finite commutative groups are uninteresting and (even, often especially) finite non-commutative groups are interesting.  
