# Principled way to find a shape with symmetries given by a group

Recently I've learned how groups correspond to symmetries of objects so I've been trying to find shapes corresponding to groups that I know (all with finite groups).

For example, I know $\mathbb Z / p \mathbb Z$ for $p$ prime is cyclic so it makes sense to try a prime-sided regular polygon because it has no reflectional symmetries, so there's nothing to do except rotate it by one turn until you've done $p$ rotations and are back where you started. But for non-cyclic groups I don't know in general how to proceed.

My problem is that so far I've basically been guessing and checking. Like I figured out a shape that has $K_4$ as its symmetries, but I wasn't actually looking to find $K_4$, I was trying to find a different group. How can I do a better job of producing a shape that I want?

An example: $\mathbb Z / 4 \mathbb Z$. I can't think of what to use. An isosceles triangle seems to correspond to $\mathbb Z / 2 \mathbb Z$ since the only symmetry is reflection along the line bisecting the non-congruent angle. So I tried gluing two isosceles triangles together along the non-congruent side, but that's how I ended up with $K_4$. And the other shapes I've tried, like a square or 6 pointed star, all have too much symmetry.

Is there even a single shape giving $\mathbb Z / 4 \mathbb Z$? Is this a general thing for $\mathbb Z / n \mathbb Z$ when $n$ is not prime? I know this is connected to the dihedral group -- what does that mean for my endeavor? Am I only able to find such a shape if the group in question is a dihedral group?

Sorry if this has been asked before, I searched and didn't see it but clearly I'm pretty new to group theory so i might have not recognized a solution when I saw it. Thanks for any help.

As an example of the kinds of shapes I want, I basically want polygons.

This is $K_4$ taken from the wikipedia article on it. I'm very happy with this shape.

For $\mathbb Z / 5 \mathbb Z$, there's the regular pentagon. That works great for me.

That sort of shape. I have been thinking very much in terms of "classical" shapes although I'd definitely be interested in broadening my "shape" horizons if that's insightful.

• Wait, why do prime-sided regular polygons have no reflectional symmetries? – angryavian Mar 16 '18 at 16:50
• What exactly do you mean by "shape"? An arbitrary mathematical object? A subset of $\mathbb{R}^n$? A simplicial complex? – Nate Mar 16 '18 at 16:54
• @Nate i've updated with an actual shape – alfalfa Mar 16 '18 at 16:58
• @alfalfa If you choose to ignore the reflectional symmetries, then the rotations of any $n$-sided regular polygon form $\mathbb{Z}/n\mathbb{Z}$. For each $n$, this is a subgroup of the full group of symmetries (rotations and reflections) of the $n$-sided regular polygon, called the dihedral group. – angryavian Mar 16 '18 at 17:03
• FYI: The idea of interpreting group elements as geometric isometries is part of "representation theory". – Blue Mar 16 '18 at 17:18