# Methods to prove that a subset generates the whole group

What kind of methods exist to prove that a subset of a finitely generated abelian group G generates G

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How is the group specified (e.g. matrices, words with rewrite rules, operators on a space, etc.)? –  Victor Miller Nov 17 '10 at 19:30
If you know the generators, represent each of the elements of your set as a linear combination of generators. Form a (non-square) matrix where row number $i$ consists of coefficients of element number $i$ in your set. Perform integer Gauss elimination procedure (i.e. you are allowed switching two rows, and subtracting/adding one row from/to another row). Eventually you will get a matrix in the row echelon form. Look how many 1's you have on the diagonal. If the number of 1's is the same as the number of generators, your set generates the whole group. Otherwise the answer is "no".