[Math] Questions on Symmetric group of degree 5

group-theory

Consider $S_5$, the symmetric group of degree five. Does it have a subgroup isomorphic to
$C_5 \times C_5$? Does it have elements of order 6? Does it have a subgroup isomorphic to $D_5$? What about a subgroup isomorphic to $D_6$?

Is there an actual method to 'working out' this question or am I just expected to look up the answer and write yes or no for each part? I looked it up here – http://groupprops.subwiki.org/wiki/Subgroup_structure_of_symmetric_group:S5 – and see that it does have an element of order 6, namely $(1,2,3)(4,5)$. In the subgroup section it doesn't mention anything about subgroups isomorphic to $C_5\times C_5$, $D_5$ or $D_6$ so I take it doesn't have subgroups isomorphic to those groups?

And again, just to clarify, is there a method to working this question out or am I correct just to look it up?

Best Answer

Mathematics should be present rather in your head than in the books only. Yes, we have method: thinking and drawing and writing.

Number five is not that big, you can think over the possibilities by yourself instead of looking up the solution for the given exercise. If you have doubts or you get stucked on your way, then we willingly help.

So, some hints:

  1. $|C_5\times C_5|=?$, does it divide $|S_5|$ at all?
  2. Can $D_5$ be fully described by the permutations of the vertices of the regular pentagon?
  3. $D_6$ is a bit trickier.. it has an element (rotation by $60^\circ$) which is the product of 2 reflections, and any reflection $r$ has order $2$ (that is, $r=r^{-1}$).
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