I am seeking to understand Dummit and Foote's chapter on Representation Theory and Character Theory (see pages 840-843 attached).

On Page 840 D&F write: (see attachment)

Suppose we are given an FG-Module V.

We obtain an associated vector space over F and a representation of G as follows:

Since V is an FG-Module, it is an F-Module i.e. it is a vector space over F.

Also for each g G we obtain a map from V to V, denoted by , defined by

for all

where is the given action of the ring element g on the element of V.

Since the elements of F commute with each it follows by the axioms of a module that for all and all we have:

that is, for each is a linear transformation.

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My problem with the above text is

1. I do not see why we can assume that " the elements of F commute with each "

2 Hence I do not follow the assertion above that

Can someone please show me explicitly how this is the case?

Would appreciate some help?

Peter