https://groupprops.subwiki.org/w/index.php?action=history&feed=atom&title=Finitely_generated_parafree_group 2026-11-02T11:26:08Z Revision history for this page on the wiki MediaWiki 1.41.2 https://groupprops.subwiki.org/w/index.php?title=Finitely_generated_parafree_group&diff=47335&oldid=prev 2013-06-28T02:31:00Z

<p>Created page with &quot;{{group property conjunction|finitely generated group|parafree group}} ==Definition== A <a href="/wiki/Group" title="Group">group</a> is termed a &#039;&#039;&#039;finitely generated parafree group&#039;&#039;&#039; if it is both a finit...&quot;</p> <p><b>New page</b></p><div>{{group property conjunction|finitely generated group|parafree group}}<br /> <br /> ==Definition==<br /> <br /> A [[group]] is termed a &#039;&#039;&#039;finitely generated parafree group&#039;&#039;&#039; if it is both a [[finitely generated group]] and a [[parafree group]]. Note that the free group that this finitely generated parafree group has the same lower central series as is a [[finitely generated free group]], and further, its rank is determined by the isomorphism type of the parafree group (explicitly, it equals the rank of the abelianization of the parafree group).<br /> <br /> ==Relation with other properties==<br /> <br /> ===Stronger properties===<br /> <br /> {| class=&quot;sortable&quot; border=&quot;1&quot;<br /> ! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions<br /> |-<br /> | [[Weaker than::finitely generated free group]] || || immediate || [[finitely generated and parafree not implies free]] || {{intermediate notions short|finitely generated parafree group|finitely generated free group}}<br /> |}<br /> <br /> ===Weaker properties===<br /> <br /> {| class=&quot;sortable&quot; border=&quot;1&quot;<br /> ! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions<br /> |-<br /> | [[Stronger than::parafree group]] || || || || {{intermediate notions short|parafree group|finitely generated parafree group}}<br /> |-<br /> | [[Stronger than::residually nilpotent group]] || || || || {{intermediate notions short|residually nilpotent group|finitely generated parafree group}}<br /> |-<br /> | [[Stronger than::finitely generated residually nilpotent group]] || || || || {{intermediate notions short|finitely generated reisdually nilpotent group|finitely generated parafree group}}<br /> |}</div>

Vipul