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 $\displaystyle \in $ G we obtain a map from V to V, denoted by $\displaystyle \phi (g) $, defined by

$\displaystyle \phi (g) (v) = g \cdot v $ for all $\displaystyle v \in V $

where $\displaystyle g \cdot v $ is the given action of the ring element g on the element of V.

Since the elements of F commute with each $\displaystyle g \in G $ it follows by the axioms of a module that for all $\displaystyle v, w \in V $ and all $\displaystyle \alpha , \beta \in F $ we have:

$\displaystyle \phi (g) ( \alpha v + \beta w ) = g \cdot ( \alpha v + \beta w )$

$\displaystyle = g \cdot ( \alpha v ) + g \cdot ( \beta w ) $

$\displaystyle = \alpha ( g \cdot v ) + \beta ( g \cdot w ) $

$\displaystyle = \alpha \phi (g) (v) + \beta \phi (g) (w) $

that is, for each $\displaystyle g \in G , \phi (g) $ 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 $\displaystyle g \in G $"

2 Hence I do not follow the assertion above that

$\displaystyle = g \cdot ( \alpha v ) + g \cdot ( \beta w )$ $\displaystyle = \alpha ( g \cdot v ) + \beta ( g \cdot w ) $

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

Would appreciate some help?

Peter