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  "body": "\n## Definition\n\nLet $A$ be an [[abelian group|082GHNPJ53]], written additively.\nA subset $X \\subseteq A$ is *linearly independent* if for all distinct $x_1, \\ldots, x_k \\in X$ and integers $n_1, \\ldots, n_k$,\n$$n_1 x_1 + \\cdots + n_k x_k = 0 \\implies n_1 = \\cdots = n_k = 0.$$\nThe *rank* of $A$ is the [[cardinality|0EBACQFAZW]] of a maximal linearly independent subset of $A$.\nIt does not depend on the choice of the maximal subset.\n\n## Remarks\n\nExample: $A = \\Z^2 \\oplus \\Z/3\\Z$ has rank $2$.\nThe set $\\set{(1, 0, 0), (0, 1, 0)}$ is independent, and it is maximal: for any $y \\in A$, $3y$ lies in $\\Z^2 \\oplus 0$, so $3y = m(1,0,0) + n(0,1,0)$ gives a relation with nonzero coefficient on $y$.\n\nNon-example of independence: in $\\Z/6\\Z$, the set $\\set{1}$ is not independent, since $6 \\cdot 1 = 0$ with coefficient $6 \\neq 0$.\nIn fact a group in which every element has finite order, such as the [[torsion subgroup|0DBWXQRXQJ]] of any abelian group, has rank $0$.\n\nEquivalently, the rank of $A$ is the [[dimension|03GSZ8RTZJ]] of the $\\Q$-vector space $A \\otimes_\\Z \\Q$.\nRank is additive: if $0 \\to A \\to B \\to C \\to 0$ is exact, then $\\rank B = \\rank A + \\rank C$.\nThe [[free abelian group|0WJX8MGG37]] $\\Z^n$ has rank $n$, and $\\Q$ has rank $1$.\n",
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