| Summary: | <p>Let <em>F</em> be the free product of <em>n</em> groups and let <em>R</em> be a normal subgroup generated (as a normal subgroup) by <em>m</em> elements of <em>F</em>, where <em>m</em> < <em>n</em>. The Main Theorem gives sufficient conditions for families of fewer than <em>n−m</em> subgroups in certain quotients of <em>F/R</em> to generate their free product. This leads to a more direct proof of a result of the first author, that if <em>G</em> is a group having a presentation with <em>n</em> generators and <em>m</em> relators, where <em>m</em> < <em>n</em>, then any generating set for <em>G</em> contains <em>n−m</em> elements that freely generate a free subgroup of <em>G</em>. Another consequence is that an <em>n</em>-generator one-relator group cannot be generated by fewer than <em>n</em>−1 subgroups each having a non-trivial abelian normal subgroup.</p>
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