Examples of groups

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Some elementary examples of groups in mathematics are given on Group (mathematics). Further examples are listed here.

Contents

Permutations of a set of three elements

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) = e; (ab)(ba) = (ba)(ab) = e; so every element has an inverse.

By inspection, we can determine associativity and closure; note in particular that (ba)b = aba = b(ab).

Since it is built up from the basic operations a and b, we say that the set {a,b} generates this group. The group, called the symmetric group S3, has order 6, and is non-abelian (since, for example, abba).

The group of translations of the plane

A translation of the plane is a rigid movement of every point of the plane for a certain distance in a certain direction. For instance "move in the North-East direction for 2 miles" is a translation of the plane. If you have two such translations a and b, they can be composed to form a new translation ab as follows: first follow the prescription of b, then that of a. For instance, if

and

then

(see Pythagorean theorem for why this is so, geometrically).

The set of all translations of the plane with composition as operation forms a group:

This is an Abelian group and our first (nondiscrete) example of a Lie group: a group which is also a manifold.

The symmetry group of a square - dihedral group of order 8

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