# The symmetries of a pentagram

## 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?

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

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?

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.

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?

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.