Can anyone help me with this.

i) Let $H$ be a subgroup of $D_{10}$, where $D_{10}$ is the Dihedral group of order 10, and $H\ne D_{10}$. Prove that $H$ is cyclic.

ii) Let $H=<\sigma >$, a cyclic subgroup of $D_{10}$ generated by a reflection $\sigma$. Find elements $x_1$, ..., $x_5$ of $D_{10}$ such that $H{x}_1$, ..., $H{x}_5$ are the five different right cosets of $H$ in $D_{10}$.

Thanks
James

Demetres

Looking at the elements of $D_{10}$ the only possible subgroups are $\{e\}$, $\{e,{\sigma }_i\}$ where $1\le i\le 5$ and finally $\{e,\rho ,{\rho }^2,{\rho }^3,{\rho }^4\}$.

Obviously these seven subgroups are all cyclic.

I am not sure how to do ii).

I know that $<\sigma >=\{e,\sigma \}$ but I can't seem to find the right cosets and the elements $x_1$, ..., $x_5$.

Regards
James Lobo

$x_1=\rho$

$x_2={\rho }^2$

$x_3={\rho }^3$

$x_4={\rho }^4$

$x_5=\sigma$ 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