- #1
OMM!
- 15
- 0
There is a Theorem that says FG-Modules are equivalent to group representations:
"(1) If [itex]\rho[/itex] is a representation of G over F and V = [itex]F^{n}[/itex], then V becomes an FG-Module if we define multiplication vg by: vg = v(g[itex]\rho[/itex]), for all v in V, g in G.
(2) If V is an FG-Module and B a basis of V, then [itex]\rho[/itex]: g [itex]\mapsto[/itex] [itex][g]_{B}[/itex] is a representation of G over F, for all g in G"
I've been told and I have read that using FG-Modules is advantageous to using group representations, but what exactly is the advantage of this, other than getting results like Maschke's Theorem?!
Thanks for any help!
"(1) If [itex]\rho[/itex] is a representation of G over F and V = [itex]F^{n}[/itex], then V becomes an FG-Module if we define multiplication vg by: vg = v(g[itex]\rho[/itex]), for all v in V, g in G.
(2) If V is an FG-Module and B a basis of V, then [itex]\rho[/itex]: g [itex]\mapsto[/itex] [itex][g]_{B}[/itex] is a representation of G over F, for all g in G"
I've been told and I have read that using FG-Modules is advantageous to using group representations, but what exactly is the advantage of this, other than getting results like Maschke's Theorem?!
Thanks for any help!