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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?

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If you have been given this as an exercise, you are certainly not supposed to look up the answer somewhere. To do this sort of exercise, think a bit about what the order of some element of $S_5$ is, given its decomposition into disjoint cycles. Also consider what sort of element orders it takes to get those mentioned subgroups, and how many such elements. And no, just because it is not mentioned, it might still have a subgroup isomorphic to one of those (it does!). –  Tobias Kildetoft Oct 25 '12 at 8:52
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I think the point of the question is to get you thinking about the structure of the groups you are working with. For example, what is the order of $C_5\times C_5$? Do you know generators and relations which define the dihedral groups - then look for elements in $S_5$ which satisfy the relations - what would they have to look like, or why can't they exist (using the properties you know)? –  Mark Bennet Oct 25 '12 at 8:56
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1 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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For #3 you can also look inside D6 for low index subgroups. This can be easier when the numbers are larger: dihedral groups don't get appreciably more complicated, but symmetric groups get more types of order 2 elements. –  Jack Schmidt Oct 25 '12 at 12:55
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