James Lobo
Posted on Thursday, 13 November, 2003 - 11:27 am:
Hi,

Can anyone help me with this.

i) Let H be a subgroup of D10, where D10 is the Dihedral group of order 10, and H ¹ D10. Prove that H is cyclic.

ii) Let H= < s > , a cyclic subgroup of D10 generated by a reflection s. Find elements x1, ..., x5 of D10 such that H x1, ..., H x5 are the five different right cosets of H in D10.

Thanks
James

Demetres Christofides
Posted on Thursday, 13 November, 2003 - 12:10 pm:
(i) Which are the only possibilities for the order of H?

(ii) H x=H y iff x y-1=1 or s. Think geometrically.

Demetres

James Lobo
Posted on Monday, 17 November, 2003 - 11:36 am:
Hi, I think I can do part i).

Looking at the elements of D10 the only possible subgroups are {e}, {e, si} where 1 £ i £ 5 and finally {e,r,r2,r3, r4}.

Obviously these seven subgroups are all cyclic.

I am not sure how to do ii).

I know that < s > = {e,s} but I can't seem to find the right cosets and the elements x1, ..., x5.

Regards
James Lobo

James Lobo
Posted on Tuesday, 18 November, 2003 - 12:13 pm:
I think that for ii) x1, ..., x5 are

x1=r

x2=r2

x3=r3

x4=r4

x5=s or e

Can someone tell me whether my answer to ii) is correct and also my previous posted message for the answer to i)

Regards
James

Demetres Christofides
Posted on Tuesday, 18 November, 2003 - 02:11 pm:
Your answers are correct. You should note that the answer in (ii) is not unique. For example Hr = Hsr, so you could also take sr instead of r for x1 e.t.c.

Demetres