Looking for a subgroup $H$ of $D_{2019}$ with the following properties:

Question

I m looking for a subgroup $H$ of the dihedral group $(D_{2019},\circ)$ so that:

With $(D_{2019},\circ)$ the group containing as elements
$b$: a reflexion on a symmetry axis, and
$a$: a rotation of $\frac{2\pi}{2019}$ On a “2019-gon”

• $H \neq D_{2019}$
• $H$ has an element with order $673$
• $a^{102} \circ b \in H \circ a$

What I have tried so far: out of the 3rd requirement I gathered: $a^{102}\circ b = a^{103}\circ b \circ a \in H \circ a$

So I thought that $H=\{1,a^3,a^6,…,ab,a^3b,…a^{2018}b\}$

But since $a^{103} \notin H$ This cannot be it. Where did I go wrong?

Any help would be very much appreciated, thank you

Answer 1

There's anti-commutativity in dihedral groups. Here $ab=ba^{-1}.$

So $$a^{102}ba^{-1}=a^{103}b.$$

So, take $H=\langle a^3,a^{103}b\rangle.$

Then $H=G$, because $a^{103}b\in G\setminus \langle a^3\rangle $.

Thus there's no way to.