Why Does the Field Choice Affect Maschke's Theorem Application to Group C2?

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catcherintherye
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In the statement of Maschke's theroem we are told 'Let G be a finite group and F a field in which |G| not equal to zero. As an example we are told if our group was C2 (cyclic) then we could not have F=F2 (the field with 2 elements). I fail to see how C2 and F2 are related, surely |C2|=2 regardless of F. I do not see how changing the field changes the size of the group. What am I missing??
 

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It doesn't alter the size of the group. Nothing says that choosing the field changes the size of the group - I don't see where you got that conclusion from.

|C_2| is certainly 2, and Maschke's result holds in any field where 2 is invertible (i.e. precisely not F_2 or any extension).

Maschke's theorem, or its proof, requires that one is able to multiply through by the multiplicative inverse of |G| in F.
 
All the versions of Maschke's theorem that I am familiar with say "over a field of characteristic k where k does not divide |G|" or the equivalent "Over a field in which |G| is not a zero divisor". If your text actually says "over a field in which |G| not equal to zero"- that is very sloppy language.