Symmetry groups, compiling Caley table

[Antonnis]

Hello. I have a homework like this:

A quadrilateral Q has vertices (2, 2), (−1, 1), (−2, −2) and (1, −1).

(a) (i) Sketch the quadrilateral.
(ii) Using standard notation, write down the elements of the symmetry group S(Q) of Q, giving a brief description of the geometric effect of each symmetry on points in the plane.
(iii) Compile a Cayley table for S(Q).

(b) (i) Show that the set G = {0, 2, 4, 6} is a group under the operation +8. You should state the inverse of each element in (G, +8).
(ii) State whether (S(Q), ◦) is isomorphic to (G, +8), justifying your answer briefly.
(iii) Sketch a plane set R whose symmetry group S(R) is isomorphic to (G, +8).

In a) ii) I know how to describe reflection symmetries [S(Q)={e, q(pi/4), q(3pi/4)}] but as I know rhombus have two rotational symmetries as well. I have a problem with them. Can you help me how to describe them?

Thank you in advance.

 

[Steven G]

Hello. I have a homework like this:

A quadrilateral Q has vertices (2, 2), (−1, 1), (−2, −2) and (1, −1).

(a) (i) Sketch the quadrilateral.
(ii) Using standard notation, write down the elements of the symmetry group S(Q) of Q, giving a brief description of the geometric effect of each symmetry on points in the plane.
(iii) Compile a Cayley table for S(Q).

(b) (i) Show that the set G = {0, 2, 4, 6} is a group under the operation +8. You should state the inverse of each element in (G, +8).
(ii) State whether (S(Q), ◦) is isomorphic to (G, +8), justifying your answer briefly.
(iii) Sketch a plane set R whose symmetry group S(R) is isomorphic to (G, +8).

In a) ii) I know how to describe reflection symmetries [S(Q)={e, q(pi/4), q(3pi/4)}] but as I know rhombus have two rotational symmetries as well. I have a problem with them. Can you help me how to describe them?

Thank you in advance.

1st of all I doubt that you have homework like this but I believe you have homework exactly like this. This is fine, no problem at all.
I do have a question about what you mean (ie your teacher/text) when you write by a group under the operation +8. Do you mean mod8?? A simple way to see if there is an isom. is to see if the Cayley tables are the same (up to names of course)

 

[Limax]

In a) ii) I know how to describe reflection symmetries [S(Q)={e, q(pi/4), q(3pi/4)}] but as I know rhombus have two rotational symmetries as well. I have a problem with them. Can you help me how to describe them?

\(\displaystyle e\) is not a reflectional symmetry.

Hint: \(\displaystyle (G,+_8)\) is a cyclic group. \(\displaystyle S(Q)\) is clearly not cyclic because it has no element of order 4. A plane figure whose symmetry group is cyclic of order 4 clearly cannot have reflectional symmetries. Can you sketch a plane figure with 4 rotational symmetries but no reflectional ones? (Hint: It can't be a quadrilateral.)