Chain Conditions for Subgroups of Infinite Order or Index

✍ Scribed by Dae Hyun Paek

• Publisher: Elsevier Science
• Year: 2002
• Tongue: English
• Weight: 115 KB
• Volume: 249
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1006/jabr.2001.9085

✦ Synopsis

A group G is said to satisfy max-∞ if each nonempty set of infinite subgroups of G has a maximal element. A group G is said to satisfy min-∞ if each nonempty set of subgroups of G with infinite index has at least one minimal element.

Groups with max-∞ or min-∞ are the subject of this paper. Abelian, nilpotent, and solvable groups with max-∞ or min-∞ are examined in detail and structure theorems are given in each case. We then characterize the groups with max-∞ or min-∞ in the smallest class of groups containing all linear groups which is locally closed and closed with respect to the formation of ascending series with factors in the class.  2002 Elsevier Science (USA)