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Determine subgroup $\langle (12),(13)(24)\rangle $ of group $ S_{4}$.

Using the definition of generator of group, I believe I am supposed to find all permutations that can be written as multiple of $(12),(13)(24)$ and/or their inverses. How do I know I where to stop, because there is a lot of elements to multiply ( for example (12),(13)(24),(1324),(12)(34),(1432) etc. )?

Shaun
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user560461
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    The first time you meet this kind of an exercise it is probably for the best to keep multiplying the already found elements by either of the generators from left or right. Do observe that both generators, call the $s_1$ and $s_2$ have order two. So if you got a "new" element $x=as_i$, then there is no need to compute $xs_i$ for it will give you back $(as_i)s_i=as_i^2=a$. – Jyrki Lahtonen Aug 16 '19 at 09:22
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    When you become more familiar with groups in general and $S_4$ in particular, then you learn about many "standard" subgroups, such as $D_4$, the group of symmetries of a square (cornered number clockwise 1,3,2,4). You will then see that both your generators are in that subgroup, so whatever they generate cannot go outside $D_4$. This will allow you to stop a bit earlier with confidence. – Jyrki Lahtonen Aug 16 '19 at 09:25

2 Answers2

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$G = \langle (12),(13)(24)\rangle = \langle (12),(12)(13)(24)\rangle = \langle (12),(3241)\rangle$. Let $s = (12)$ and $r = (3241)$ we have $G = \langle r,s\mid r^4 = s^2 = 1, srs = r^{-1}\rangle\cong D_8$, where $D_8$ denotes the dihedral group of order $8$, and $G = \{1,r,r^2,r^3,s,sr,sr^2,sr^3\}$.

Groups
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Yes, this is D4, the dihedral group on four letters, i.e. rotations and a flip of a square on a plane.

For the given notation we number the four vertices clockwise 1,4,2,3. We designate the first generator (1,2) as $g_1$ to be the flip over the main diagonal, which joins vertices 1 and 2. The second generator is a flip fixing the two lateral sides.

Clockwise rotation by 90 degrees is applying the second generator, then the first one.

To quickly check oneself, go to Online Magma Calculator. Paste the following stuff

K := PermutationGroup< 4 | (1,2), (1,3)(2,4) >; K; Order(K); IdentifyGroup(K); GroupName(SmallGroup(8,3));

It says it is D4. In the above <8,3> means nothing special, group listed third of groups of order 8 in the Magma database.

Again, I put references to Magma not because I promote it, but it is a quick and dirty way to solve this type of problems.

Mikhail D
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