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Free group
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<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the free group <i>F</i><i>S</i> over a given set <i>S</i> consists of all <a href="/facts/Word_(group_theory)/u9jJeb9w">words</a> that can be built from members of <i>S</i>, considering two words to be different unless their equality follows from the <a href="/facts/Group_(mathematics)/IQTYqINt">group axioms</a> (e.g. <i>st</i> = <i>suu</i>−1<i>t</i> but <i>s</i> ≠ <i>t</i>−1 for <i>s</i>,<i>t</i>,<i>u</i> ∈ <i>S</i>). The members of <i>S</i> are called generators of <i>F</i><i>S</i>, and the number of generators is the rank of the free group. An arbitrary <a href="/facts/Group_(mathematics)/IQTYqINt">group</a> <i>G</i> is called free if it is <a href="/facts/Group_isomorphism/LC4N1aDw">isomorphic</a> to <i>F</i><i>S</i> for some <a href="/facts/Subset/SJGERmbA">subset</a> <i>S</i> of <i>G</i>, that is, if there is a subset <i>S</i> of <i>G</i> such that every element of <i>G</i> can be written in exactly one way as a product of finitely many elements of <i>S</i> and their inverses (disregarding trivial variations such as <i>st</i> = <i>suu</i>−1<i>t</i>). </p><p>A related but different notion is a <a href="/facts/Free_abelian_group/oaSRGmpY">free abelian group</a>; both notions are particular instances of a <a href="/facts/Free_object/VxL28y8B">free object</a> from <a href="/facts/Universal_algebra/uMJAL6ER">universal algebra</a>. As such, free groups are defined by their universal property. </p>