This question is from Gallian Page 69 Q 9. Suppose a subgroup od D_4 contains H and D. I want to show that these two generated whole of $D_4$. Now rotation will generate other rotations which is fairly clear. But i am confused about how reflections can generate other reflections? Because a reflection say V will, if applied again and again can generate only V and identity which is $R_0$ but how do other reflections can be generated by this
Thanks a lot
The dihedral group can be said to have two generators. That is you can find two elements of the set of group elements which by iterated operation in different orders can create any other member of the set. Compare to the simpler cyclic group where it is enough with 1 generating element. $C_3$ Rotation 120 degrees once, twice and thrice "generates" all three group elements. That makes the set {rotation by 120 degrees} a generating set of size 1 for $C_3$.
For $D_4$ you can pick one particular reflection and say one of the minimal angle rotations and they will together be a generating set. But you may need to build sequences involving both of these generators to generate all elements of the group. You can read more for example here generating set at Wikipedia.