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

Question

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. )?

Answer 1

$G = \langle (12),(13)(24)\rangle = \langle (12),(12)(13)(24)\rangle = \langle (12),(3241)\rangle$. 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 $G = \{1,r,r^2,r^3,s,sr,sr^2,sr^3\}$.

Answer 2

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.