the symmetry group of a figure
A figure is symmetric when there is some way to move it that lands it back exactly on itself — turn a square a quarter turn, or flip a butterfly across its middle, and you cannot tell it was touched. The symmetry group of a figure simply collects all such self-matching motions into one set. It is the complete catalogue of every rigid motion that leaves the figure looking unchanged, and that catalogue is a remarkably faithful fingerprint of how symmetric the figure really is.
Precisely, the symmetry group of a figure F is the set of all isometries that map F onto itself. This set always contains the do-nothing identity (which trivially fixes F), and it is closed and reversible: if two motions each preserve F, so does doing one then the other, and so does undoing either — these are exactly the properties that make it a group under composition. For a non-square rectangle the symmetries are: the identity, a 180-degree rotation, a horizontal flip, and a vertical flip — four in all. For a square there are eight: four rotations (by 0, 90, 180, 270 degrees) and four reflections (two through opposite-side midpoints, two through opposite corners). The number of elements is the 'order' of the group and measures the total amount of symmetry.
Symmetry groups let us compare and classify shapes by their symmetry rather than their size or position. A figure with only rotations has a cyclic symmetry group; one with both rotations and reflections has a dihedral group; the orbit of a point under the group is the set of all places that point gets sent. This is the bridge from geometry into the language of groups, and it scales all the way up to the seven frieze groups and seventeen wallpaper groups, which are nothing but the symmetry groups of repeating infinite patterns.
List the symmetries of a (non-square) rectangle. Identity: leave it alone. 180-degree rotation about its centre: corners swap diagonally, fits back on itself. Reflection across the horizontal centre line: top and bottom swap. Reflection across the vertical centre line: left and right swap. A 90-degree turn does NOT work (it would stand the rectangle on end). So the symmetry group has exactly four elements.
Collect every motion that maps the figure to itself; counting them measures how symmetric it is.
The identity (doing nothing) always belongs to the symmetry group, so even a totally lopsided shape has a symmetry group — of order one. 'No symmetry' means the group contains only the identity, not that the group is empty.