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How to find a base of the permutation group G=⟨x,y⟩≤S4 x=(1,2,3), y=(1,2,4)?

I hear that base for G is a sequence B = [$b_1, ..., b_m$] ⊂ Ω such that the only element of G which stabilizes each $b_i$ is the identity

I am new to Group theory and I read somewhere that the <> notation is for generators so in this case it will be the list of generators formed by the cartesian product as listed below:

<1,1>, <1,2>, <1,4> , <2,1>, <2,2>, <2,4>, <3,1>, <3,2> and <3,4>

And each of these generators will generate a set so should I compute all these generating sets and make sure none of these sets (besides the identity set) stabilizes each $ b_i $ of the base?

Olexandr Konovalov
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user1870400
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  • This is the example described in http://math.stackexchange.com/questions/1662723 The example illustrates the computation of the base, which is $[1,2]$. I cannot make any sense of your question. – Derek Holt Feb 22 '16 at 09:03
  • I can't thank you enough! but if I may ask can you please explain me how you arrived these two steps? "Start with a single base point 1, and (strong) generating set S=[x,y].

    The orbit of 1 is O1=[1,2,3,4] with permutations T1=[1,x,x−1,xy] mapping 1 to the orbit points."

     – user1870400 Feb 22 '16 at 10:12
  • other words here are my questions 1) How did you think that SGS is of two elements x,y ? why not three elements x,y,z or one element x or y 2) how did you know 1 is the base point? I am sure I am missing something and I am really sorry if it is too naive for you but I sincerely hope you don't mind asking me questions. Again I can't imagine someone of you stature answering my questions which may look silly initially because I am new to group theory however I am trying really hard to parse through the notation. Concrete examples always help me! – user1870400 Feb 22 '16 at 10:18
  • These are just the initialization steps of the algorithm. $x$ and $y$ were the two given (i.e. input) generators. So we start with those as the provisional strong generators. We expect to have to adjoin new strong generators during the course of the algorithm. Choosing $1$ as the first base point is again just the first step of the algorithm. Again we expect to have to adjoin new base points later. – Derek Holt Feb 22 '16 at 10:55
  • will there always be two given generators for any group? after reading some more literature what does this notation me G = <(1,2,3), (1,2,4)> ? Does this mean the group has only two elements ? – user1870400 Feb 22 '16 at 11:27
  • The group could have any number of input generators, and these will form the initial strong generating set. The notation just means that $G$ is generated by $(1,2,3)$ and $(1,2,4)$. The number of elements in the group is what the algorithm calculates which, as we saw in the example, is $12$. – Derek Holt Feb 22 '16 at 11:34

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