Concept

# Examples of groups

Summary
Some elementary examples of groups in mathematics are given on Group (mathematics). Further examples are listed here. Permutations of a set of three elements Dihedral group of order 6 Consider three colored blocks (red, green, and blue), initially placed in the order RGB. Let a be the operation "swap the first block and the second block", and b be the operation "swap the second block and the third block". We can write xy for the operation "first do y, then do x"; so that ab is the operation RGB → RBG → BRG, which could be described as "move the first two blocks one position to the right and put the third block into the first position". If we write e for "leave the blocks as they are" (the identity operation), then we can write the six permutations of the three blocks as follows:
• e : RGB → RGB
• a : RGB → GRB
• b : RGB → RBG
• ab : RGB → BRG
• ba : RGB → GBR
• aba : RGB → BGR
Note that aa has the effect RGB → GRB → RGB; so we can write aa = e. Similarly, bb = (aba)(aba
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