Editing Associated bundle (section)

===Principal bundle associated with a fibre bundle===
As before, suppose that <math>E</math> is a fibre bundle with structure group <math>G</math>.  In the special case when <math>G</math> has a [[Group action (mathematics)#Types of actions|free and transitive]] left action on <math>F'</math>, so that <math>F'</math> is a principal homogeneous space for the left action of <math>G</math> on itself, then the associated bundle <math>E'</math> is called the principal <math>G</math>-bundle associated with the fibre bundle <math>E</math>. If, moreover, the new fibre <math>F'</math> is identified with <math>G</math> (so that <math>F'</math> inherits a right action of <math>G</math> as well as a left action), then the right action of <math>G</math> on <math>F'</math> induces a right action of <math>G</math> on <math>E'</math>. With this choice of identification, <math>E'</math> becomes a principal bundle in the usual sense. Note that, although there is no canonical way to specify a right action on a principal homogeneous space for <math>G</math>, any two such actions will yield principal bundles which have the same underlying fibre bundle with structure group <math>G</math> (since this comes from the left action of <math>G</math>), and isomorphic as <math>G</math>-spaces in the sense that there is a <math>G</math>-equivariant isomorphism of bundles relating the two.

In this way, a principal <math>G</math>-bundle equipped with a right action is often thought of as part of the data specifying a fibre bundle with structure group <math>G</math>, since to a fibre bundle one may construct the principal bundle via the associated bundle construction. One may then, as in the next section, go the other way around and derive any fibre bundle by using a fibre product.