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I'm not very familiar with group theory, so I'm sorry if this is obvious, but I'm trying to find a specific group so I can see its isomorphisms.

If you take a set of four pairs of two elements, for example $\{(1,2),(3,4),(5,6),(7,8)\}$, swapping the elements of any pair constitutes an action. In total, there's $2^4=16$ actions in all. So for example, this permutation $$ \begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ 1 & 2 & 4 & 3 & 6 & 5 & 7 & 8 \end{pmatrix} $$ is a part of the group, but $$ \begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ 1 & 3 & 2 & 6 & 4 & 5 & 7 & 8 \end{pmatrix} $$ is not.

I've been looking around for ways to figure this out but I can't find anything. WolframAlpha doesn't seem to have any group-describing functions from what I've found, and all the lists of groups only seem to list them up to isomorphism, which is particularly frustrating. So what is this group called? And furthermore, if I had any set of permutations that met the definition of a group, how could I figure out what group it was?

Edit: reworded some things to be a little clearer, hopefully.

krytton
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  • It's not clear to me what you're asking. Why isn't that permutation valid? – Shaun Sep 24 '21 at 12:49
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    @Shaun because ti swaps group of size 3, not size 2. krytton, I don't understand what the group operation would be you are looking for? – gt6989b Sep 24 '21 at 12:50
  • Do you want to ask if the given permutations form a subgroup of all permutations of the $8$ element set? – Berci Sep 24 '21 at 12:53
  • @gt sorry it was unclear, each operation swaps some number of pairs. A physical analogy would be: if you had four cards, you could flip any of them over, but you couldn't change what order they were in. – krytton Sep 24 '21 at 12:58
  • Each permutation is a product of transpositions. – Shaun Sep 24 '21 at 13:02
  • In light of the above, I still don't understand the question. – Shaun Sep 24 '21 at 13:04
  • Okay, so in that sense, the group consists of all the possible products of {(1,2),(3,4),(5,6),(7,8)}. Does that make more sense? – krytton Sep 24 '21 at 13:07
  • What excludes the likes of $(23)$? – Shaun Sep 24 '21 at 13:09
  • $(2,3)$ wasn't one of the pairs. I know it wasn't clear in the original question but we split the original list into four pairs, and then generate the permutations from those specific pairs. – krytton Sep 24 '21 at 13:13
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    So your question is something like "I have a group. How do I find out what group it is?" What kind of answer are you looking for? The group in your example is elementary abelian of order $2^4$, which can be written in many ways, such as $C_2^4$, ${\mathbb Z}_2^4$ (I don't like that one because ${\mathbb Z}_2$ has too many different meanings), $\mathtt{SmallGroup}(16,14)$, or even (in ATLAS notation) just $2^4$. – Derek Holt Sep 24 '21 at 13:47
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    Alternatively, perhaps you are asking how to study a group defined as the group generated by a collection of (finite) permutations. You can start by running the Schreier-Sims algorithm, which will tell you the order of the group. – Derek Holt Sep 24 '21 at 13:48
  • @DerekHolt I was asking both, basically, and that answers both of them, so thanks! That really helps. – krytton Sep 24 '21 at 13:57

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