Hello, this is my first post.

I was wondering how, if we have a finite group and a field and an -module , how we would show there is no -submodule complementary to ? Obviously a proof by contradiction but how would I start?

Thanks,

John

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- October 11th 2009, 11:15 PMJohnnyFrankstonFG-modules
Hello, this is my first post.

I was wondering how, if we have a finite group and a field and an -module , how we would show there is no -submodule complementary to ? Obviously a proof by contradiction but how would I start?

Thanks,

John - October 11th 2009, 11:44 PMNonCommAlg
- October 11th 2009, 11:46 PMJohnnyFrankston
- October 12th 2009, 02:49 AMNonCommAlg
suppose and let see that for all and thus for all we have for some and so now suppose that

then since are submodules, we'll have which is nonsense. hence which means that as an element of is non-zero. Q.E.D.

: note that is an ideal of i.e. it's both left and right sumodule. so if we assume that is a right submodule and then - October 12th 2009, 03:34 AMJohnnyFrankston
- October 12th 2009, 03:50 AMNonCommAlg