Proving a group G is isomorphic to D_10

by alex07966
Tags: dihedral, group, isomorphic, proof, prove
alex07966 is offline
Nov2-09, 07:31 AM
P: 7
Question: Let G be a non-abelian group such that the size of G is 10. Prove G is isomorphic to D_10.

I have started by saying the elements of G have order 1,2,5 or 10. And then showing how there are no elements of order 10 as that would make it abelian. I then show there is an element x of order 5 and an element y of order 2.
I then show G = {e, x, x^2, x^3, x^4, y, xy, (x^2)y, (x^3)y, (x^4)y} using right cosets.
NOW... i need to find what yx is equal to:
I then show that yx must be equal to either (x^2)y, (x^3)y or (x^4)y.
I think i then need to show BY CONTRADICTION that yx is not equal to the first two elements and therefore must be equal to (x^4)y. I cant reach a contradiction though!!! please help.

I can then say this group G has the same equations for working out the multiplication table of D_10 and is therefore isomorphic to D_10.
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Billy Bob
Billy Bob is offline
Nov2-09, 01:19 PM
P: 394
Here's one way. Write [tex]yxy^{-1}=x^m[/tex].

Try to obtain [tex]x=y^2 xy^{-2}=(x^m)^m[/tex] and thus [tex]m^2[/tex] is congruent to 1 modulo 5.

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