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Given a matrix whose every element is an element of group Sn (Symmetric group of n objects). I want to apply Gaussian Elimination to convert it into row echelon form. I need to find out linearly independent vectors.

What elementary matrix operations can I perform on such matrix?

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Matrices and vector spaces don't make sense over a group. I don't think there is a sensible definition of matrix multiplication, nor is there a way to add/subtract vectors from each other (to row-reduce and to define linear independence). –  dbaupp Jun 17 '12 at 8:24
Are you working in $M_\ell(k[G])$, the $\ell\times\ell$ matrices in the group algebra? –  anon Jun 17 '12 at 8:38
And what do you mean by "such a group"? A matrix is not a group!. –  Derek Holt Jun 17 '12 at 10:00
@anon: what does that symbols mean in your comment? –  Donotalo Jun 17 '12 at 12:24
@DerekHolt: Wrong wording, sorry. The elements of the matrix comes from Symmetric group. –  Donotalo Jun 17 '12 at 12:24
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