Is the Group of Symmetries of a Pentagram Isomorphic to the Dihedral Group?

[fk378]

Homework Statement

The group of symmetries of a regular pentagram is isomorphic to the dihedral group of order 10.

Show that this is true.

The Attempt at a Solution

It seems to me that the group shown by the "star" has order 5, since, by following the lines from one point, it takes 5 total paths to get back to the original point.

So I thought it would isomorphic to the cyclic group of order 5...how is it isomorphic to the dihedral group?

 

[Daettil]

The rotational symmetries are isomorphic to the cyclic group of order 5, but there are also reflectional symmetries that need to be considered.

 

[fk378]

By reflectional symmetries do you mean the inverse? As in, if I choose one path from point A to point B to point C, then the reflectional symmetry is from point C to point B to point A?

 

[Daettil]

If you center your pentagram about the origin then the reflection across the x-axis would be a reflectional symmetry. You should be able to generate your group from that reflection and the rotation of 72 degrees about the origin.

 

[fk378]

Oh okay...so when I said this---

The Attempt at a Solution

It seems to me that the group shown by the "star" has order 5, since, by following the lines from one point, it takes 5 total paths to get back to the original point.

So I thought it would isomorphic to the cyclic group of order 5...how is it isomorphic to the dihedral group?

---was I wrong?

 

[Daettil]

Yes the group will have order 10. If you label a vertex A and label a vertex adjacent to A B. Then there are 5 positions that A can be moved to and each of those allows exactly 2 positions for B.