= Maximal condition on subgroups
{title2=$\mathrm{max}$}
A <group> has the maximal condition on subgroups when every ascending sequence of <subgroups> eventually stabilizes. Equivalently, every subgroup is a <finitely generated group>. If a subgroup were not finitely generated, repeatedly adjoining an element outside the previously generated subgroup would give a strictly ascending sequence. Conversely, generators of the union of an ascending sequence all lie in one term, making that term the whole union. This condition is preserved by <group extensions> because each subgroup is an extension of its intersection with the kernel by its image in the quotient.
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