Mathematics & Groups?

Question

If we consider the set of all possible isometries of a) a square, b) a non-square rectangle (an "oblong") which keep both the centre and the overall positions of all the vertices unmoved (i.e. no translations or glide reflections).
In each of the case, how many different isometries of the shape are there? And what would happen when we combine all possible pairs of these isometries?
Also,
In each case (a) and (b), does the full set of permitted isometries transformations form a group(with respect to the operation "Tranformation 1 is followed by Transformation B")?
And if they both form groups, are the two groups equivalent (in the same way that the "addition modulo 4" and "multiplication on {1, -1, i, -i}" were equivalent)?
How can I explain this?

I would appreciate any help! Thanks

Answer 1

if you're keeping the center fixed, then you're limited to certain rotations and reflections. for the square, you can rotate by 0, 90, 180, or 270 degrees, and you can reflect across 4 different axes (horizontal, vertical, and two diagonals). so there are 8 different isometries.

a good exercise would be to make a square out of paper and somehow label the corners so you can tell them apart, and then see what happens when you combine pairs of isometries. be warned that order matters: for example, if you rotate by 90 degrees and then reflect across the horizontal axis, it's different from if you reflected first and then rotated.

for the oblong rectangle, there are only two rotations (0 and 180 degrees), and the diagonal reflections don't work anymore. so there are only 4 isometries.

in both cases, you get a group. you've basically proved that by showing that "isometry 1 followed by isometry 2" is again an isometry. the identity map is an isometry, and you can always invert an isometry (either rotate in the opposite direction or reflect again).

the two groups can't be equivalent since they don't have the same number of elements.