1. ## Regulat Octagon

Hi all

Got this question but dont know where to start - need to get this in by tonight:

The Group D8 (i.e. a regular Octagon with 8 sides) has 5 distinct sub-groups of order 4. What I need to do is find one cyclic sub group and one non-cyclic subgroup and also give their generators, or elements in standard form.

Any ideas?

Are the elements of the subgroup: {e, r, r^2, r^3, r^4, r^5, r^6, r^7, s, sr, sr^2, sr^3, sr^3, sr^4, sr^5, sr^6, sr^7}

Cheers
Muncha

2. if you could post the elements of the group, maybe i could help.. did you mean $D_8$? This is the dihedral group on 8 vertices.

3. Hi it is D8.

The elements of D8 can be written as:

4. im not sure about this.. let me recall my group theory.. you need a sub-group of order 4, which means a group with 4 elements right?

consider $\left\{e, r^4, s, sr^4\right\}$.. i think this is isomorphic to Klein-4

how about $\left< r^2 \right> = \left\{e, r^2, r^4, r^6\right\}$

5. I think that is the element of the subgroup <r^4,s>.

What I need is one cyclic and one non-cyclic sub group. Is the one you mentioned the non-cyclic group?

6. the first set i posted is isomorphic to klein-4 and take note that klein-4 is not cyclic and hence the set cannot be cyclic..

7. Thanks Kalagota

So there are no cyclic groups, only a non cyclic group.

8. the second set i posted is a cyclic sub-group generated by $\left< r^2\right>$