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I've just been introduced to the Sylow theorems in lectures and have been shown some examples of classifying groups of specific orders, e.g. 175. I'm quite interested in learning about the classification of finite groups, so my question is this: Is there a general method for classifying all groups of a given order?

The methods I have learnt in lectures do not seem especially illuminating and only work in certain cases, but I know there are many theorems about certain types of orders e.g. prime-square order, prime-cube order, product of two distinct primes, that appear that appear to use the theorems in far greater generality.

The general method I have at my disposal at the moment for classifying groups of order $n$ is as follows:

(1): Find prime factorisation of order.

(2): For each prime divisor $p$ of $n$, find all possible values of $s_p$, the number of $p$-Sylow subgroups using Sylow theorems.

(3): Use counting arguments to restrict values of $s_p$ further.

(4): This usually yields 1 or 2 normal subgroups so a direct product can be formed.

This may work for orders 15, 175 etc. but seems to need some generalisation for most cases. How can this approach be generalised?

Any assistance would be much appreciated.

user829347
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    It cannot be generalised so easily, think of groups of order $n$ having no normal subgroups except trivial ones. The classification takes some thousand pages. Or google for the classification of groups of order $n\le 2000$. – Dietrich Burde Nov 13 '19 at 20:21
  • I'm not trying to single-handedly reproduce the complete classification of finite simple groups, just looking to reasonably widen the scope of my approach outlined above for groups of relatively small order. For example, it does not even completely work for order 6, but there is a theorem for order 'product of two distinct primes' that covers this case. – user829347 Nov 13 '19 at 20:41
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    I think, $n=2000$ is reasonably small in comparison to the monster group. Did you start googling for it already? The authors present there many interesting facts you are looking for. – Dietrich Burde Nov 13 '19 at 20:42
  • Do you mean using GAP program? With order 6 for example, I know that the only groups are $C_6$ and $D_6$, but the approach I have learnt does not seem to work even for this. – user829347 Nov 13 '19 at 20:54
  • Groups of order $6$ are a nice example (google at this page, e.g., here). For $2000$ one uses GAP, yes, but also some interesting group theory generalising Sylow theorems, i.e., group actions etc. – Dietrich Burde Nov 13 '19 at 21:03

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