# Find all characteristic subgroups of the Dihedral group $D_{12}$.

In my notation $$D_{12}=\langle \rho,\tau : \rho^6=\tau^2=1,\ \rho \tau \rho=\tau\rangle$$ So firstly, I know that all characteristic subgroups are normal. Thus, the possible candidates of $D_{2n}$ for the characteristic subgroups are $$\{1\},D_{12},<\rho>,<\rho^2>,<\rho^3>,<\rho^2,\tau>,<\rho^2,\rho\tau>$$ Is there now a property or tool which I can use to pick out the characteristic subgroups. I know for sure that $D_{12}$ and $\{1\}$ as well as the commutator subgroup $D_{12}'=<\rho>$ are characteristics, but what about the others.

• Can you help me to list the elements of the subgroup of $<\rho^2,\rho \tau>$. It's all possibgle products of $\rho ^2$ and $\rho \tau$, right? – zermelovac Aug 21 '18 at 19:50
• I gor $<\rho^2,\rho \tau >=\{1,\rho^2,\rho^4,\rho \tau, \rho^3\tau, \rho^5\tau\}$ – zermelovac Aug 21 '18 at 19:57
• So, since $<\rho^2>,<\rho^3> and <\rho^2,\tau>$ have respectively orders 3,2 and 5, they are characterstics. I just need to check for the last one? – zermelovac Aug 21 '18 at 19:59
• The first two groups; yes. The group $\langle\rho^2,\tau\rangle$ does not have order 5. In general, the order of a subgroup divides the order of the group. – Servaes Aug 21 '18 at 20:00
As for the remaining $2$ subgroups, $\langle \rho^2,\tau\rangle$ and $\langle \rho^2,\rho\tau\rangle$ there is an automorphism of $D_{12}$ sending them to each other... Namely, the one determined by $\rho\to\rho$ and $\tau\to\rho\tau$. (By the relation $\rho\tau\rho=\tau$, the elements $\tau$ and $\rho\tau$ both have order $2$; or, they are both reflections.)