1. Prove cyclic

Prove that every subgroup of D_n of odd order is cyclic

2. Originally Posted by mandy123
Prove that every subgroup of D_n of odd order is cyclic
Remember that $D_n = \{ a^ib^j| 0\leq i\leq n-1, 0\leq j\leq 1, ba=a^{n-1}b\}$
Notice that $(a^kb)^2 = a^k ba^k b = a^k a^{-k} bb = b^2 = 1$
This means that each $b,ab,...,a^{n-1}b$ have order $2$.
Thus, this subgroup cannot contain these elements for then Lagrange's theorem would imply $2$ two divides its order.

This means if $H$ is a subgroup of $D_n$ of odd order it contains the elements $e,a,...,a^{n-1}$. Thus, it is a subgroup of $\left< a\right>$. But every subgroup of a cyclic group is cyclic. Thus, $H$ is cyclic itself.