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2identity 3rotation subgroup 4reflection subgroup
List the symmetries in each of the four diagrams, that is the transformations which map the picture into itself.

What transformations do you get if you combine the transformations from diagrams 3 and 4 in all possible ways?

A set of transformations, with the operation of combining transformations, becomes a group when the set includes all possible combinations of the transformations in the set.

Do the sets of transformations you have listed form groups when their transformations are combined?

Can you find any other groups of transformations which map a regular pentagon to itself?

How many distinctly different groups are there which map a regular pentagon into itself?

Note: You don't need to do more in order to submit a solution but why not pose some similar questions for yourself? The article Paint Rollers for Frieze Patterns explores in 3D the same idea of groups of symmetries.