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By Cayley's theorem every abstract group is isomorphic to some permutation group. Since the permutation group viewpoint has the advantage of considering the actions of the group on different sets, and therefore, of finding structure not just in the underlying set of the group, but in the behaviour of its elements, why do we not always consider the permutation representations of a group?

In other words, is there an advantage to looking at "groups" proper rather than groups of permutations? Is the answer different for finite and infinite groups?

(Edit) In the comments, it was pointed out that the details introduced by considering a particular permutation/linear representation may be a hinderance when, for example, the focus is on combinatorial properties of groups given as their group presentation. What are other examples of situations in group theory when the abstract view of the group is preferred?

Sveti Ivan Rilski
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  • It depends. Are you thinking about finite groups only? – Moishe Kohan Apr 15 '20 at 16:54
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    Sometimes it's less useful to get bogged down with the details of a specific permutation representation of a group and it is more useful to consider abstract generators and relations. – Matt Samuel Apr 15 '20 at 16:55
  • @MoisheKohan Both. I will edit the question to consider the difference. – Sveti Ivan Rilski Apr 15 '20 at 16:56
  • @MattSamuel I can sort of see that intuitively, but could you give me a specific example of such a situation? – Sveti Ivan Rilski Apr 15 '20 at 16:59
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    @ivansvetirilski Finite Coxeter groups have pretty good permutation representations, but for many problems we're more interested in expressing the elements as products of generators because all Coxeter groups share certain combinatorial properties for expressions of elements in terms of specific generators. Using the permutation representation there are various seemingly unrelated ways to tell that $u\leq v$ in Bruhat order, but there's a uniform way in terms of generators. The permutation representation is faster for computation, but the generality is lost. – Matt Samuel Apr 15 '20 at 17:14
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    @MattSamuel Thank you for this insight! – Sveti Ivan Rilski Apr 15 '20 at 17:21
  • On Cayley’s theorem, and on groups as permutation groups – Arturo Magidin Apr 15 '20 at 18:07
  • Sometimes it's useful to consider permutation representations of a group; sometimes it's useful to consider linear representations; sometimes it's useful to consider generators and relations; etc. – Andreas Blass Apr 15 '20 at 18:23
  • @AndreasBlass Yes, that is true. I guess I am looking for specific instances of when each representation is useful, because on first impression, it looked to me as if the detail offered by permutation/linear representations is invariably more useful. Matt Samuel's comment gave a useful example of when this impression is wrong. I have edited the question to ask for more specific examples. – Sveti Ivan Rilski Apr 15 '20 at 18:53
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    As a preface, group theory is not my area of expertise. That being said, I can't think of a single instance where I've had a group arising from a non-permutation-related situation and wanted to represent it by permutations - in general I find permutations more difficult to understand than whatever other representation naturally arises from context. The problem is, permutations are too powerful and can make the unique properties of each group less clear. – Dark Malthorp Apr 15 '20 at 19:24
  • In some sense, the whole of mathematics is just set theory. That doesn't help much, however. Consider the following problem: describe up to isomorphism the groups of order $8$. What would you do here? – the_fox Apr 15 '20 at 19:47

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Extended comment:

in history (finite) permutation groups were discovered much before groups (in modern language: set of permutations of a set including the identity and stable under composition). Note that the notion of abstract group isomorphism was already known at that time.

The transition to abstract groups was discovered around 1880, and simultaneously infinite groups were discovered. (It was not yet obvious that "group" would not eventually mean what "monoid" actually means, and I've actually seen a paper from the 1910s (by Andreoli) writing "group" for "monoid".)

The abstract definition, for finite groups, has helped in considerable progress notably by Frobenius and Burnside, on representation theory, and subsequent ones.

Some traces of this history are visible now: for instance in the pre-abstract area one had to distinguish, in a group, the number of letters, and the number of "substitutions". This is why we call the "order" of a finite group rather than its cardinal.

(In contrast infinite groups were born already with the abstract point of view, partly due to the fact that the notion of infinite set was somewhat not well-admitted.)

Since then the teaching is traditionally exclusively focussed on groups in its abstract definition, but this makes it much more abstract and one could imagine teaching otherwise (I'm not aware of modern experiments in this direction.) Indeed mathematicians of that time had difficulty with this step of conceptualization (law on an abstract set, associativity etc), so it's not surprising students do as well.

YCor
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