Ok, so bare with me here, there's quite a few questions. I am looking at this website: https://www.math.wisc.edu/~mstemper2/Math/Pinter/Chapter13F. It's basically explaining the method of going about finding all (two: $\mathbb{Z}_6$ and $S_3$) of the isomorphism classes of groups of order six.
It starts with Cauchy's Theorem: "Let $G$ be a finite group and $p$ be a prime, if $p$ divides the order of $G$, then $G$ has an element of order $p$". Using this theorem makes perfect sense to me, though I don't fully understand the proof. As a side question, if there is a relatively simple proof for Cauchy's Theorem that you could point me towards, I'd greatly appreciate it. Basically for $G=\{S,*\}$ and $|G|=6$, since the prime factorisation of $6$ is $2 \times 3$, there must be $\{e,a,b,b^2\} \subset S:(|e|=1) \land (|a|=2) \land (|b|=3)$.
The orders of $e$ (1) and $a$ (2) are different to that of $b$ (3) and $b^2$ (3). And $b \neq b^2$ because $e \neq b$ (again due to their different orders). Therefore $e \neq a \neq b \neq b^2$ and every element in $\{e,a,b,b^2\}$ is distinct. Then $ab$ and $ab^2$ are also both distinct from those four other elements since they both have an order of six (different to they other four) and $ab \neq ab^2$ because $e \neq b$. So $S=\{e,a,b,b^2,ab,ab^2\}$.
But the way they differentiate the two isomorphic classes of groups is by; one, having (i) $ba=ab$ leading to $\mathbb{Z}_6$; two, having (ii) $ba=ab^2$ leading to $S_3$. This doesn't seem intuitive to me. Why go about doing that? I get that $ba$ cannot be equal to $\{e,a,b,b^2\}$. And the only other ways in which these two element can interact without repeating themselves are $\{ba,b^2a\}$ since powers of $a$ higher than one repeat themselves and powers of $b$ higher than 2 repeat themselves. And obviously $ba \neq b^2a$ since $e \neq b$. But then what about $bab$ and $baba$ and $b^2ab^2$ and so on?
Are all of these other arrangements of $a$s and $b$s automatically closed within this group without any assumptions, or are they only closed once you assume (i) or (ii)? Essentially, is there a reason why only the relationship between $ab$, $ab^2$, $ba$, and $b^2a$ need to be considered? And why not consider all; $ab=ba$; $ab=b^2a$; $ab^2=ba$; $ab^2=b^2a$? Is it just the case that they happen to result in isomorphic classes of groups in this example, and should all of these be considered when trying to find all of the isomorphic classes of groups of a particular order? Or do these always cancel out and there's only a smaller number of cases you need to consider (since – for larger orders – my proposed method would quickly increase in cases to test)?
The same website finds all of the isomorphic classes of groups of order ten (https://www.math.wisc.edu/~mstemper2/Math/Pinter/Chapter13G) and then of order eight (https://www.math.wisc.edu/~mstemper2/Math/Pinter/Chapter13H). The website is good at showing how to get to these different isomorphic classes of groups of those particular orders, but not so good at showing why they go about the way they do (since they show you how to find them all with the prompts). I'm looking for an efficent and exhaustive general method. How could you this general method find every class and also know that there cannot be any more classes for any order group? If you could use the same genral method for order ten and eight as you'd propose for six, that'd be greatly appreciated.
I haven't been able to find it yet myself, but (maybe implies from what I've seen) if there's any half methods that are still general to find all of the isomorphism classes of abelian groups or solvable groups (whatever those are) that'd help too.
Best Answer
"I'm looking for an efficent and exhaustive general method."
Such a method does not exist, at least for now, probably ever. The number of groups of order 2048 is not known, for example. This means that soluble groups are impossible to easily classify.
Abelian groups are easy though, because of the fundamental theorem of finite abelian groups, which states that every finite abelian group is a direct product of cyclic groups.