Vol. 25, No. 2, 1968

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Transitive and fully transitive primary abelian groups

Phillip Alan Griffith

Vol. 25 (1968), No. 2, 249–254

This paper is concerned with transitivity and full transitivity of primary abelian groups. It is well known that countable primary groups and primary groups without elements of infinite height are both transitive and fully transitive. The question of whether all primary groups are transitive or fully transitive was recently answered negatively by C. Megibben. Megibben’s examples indicate that pωG may be transitive (fully transitive) while G is not transitive (fully transitive). For β an ordinal number, we investigate conditions on a primary group G which will insure that G is transitive (fully transitive) whenever pβG is transitive (fully transitive). Specifically, we show that if G∕pβG is a direct sum of countable groups and pβG is fully transitive, then G is fully transitive. The same result is established for transitivity except that β is restricted to be a countable ordinal.

Mathematical Subject Classification
Primary: 20.40
Received: 26 January 1967
Published: 1 May 1968
Phillip Alan Griffith
Department of Mathematics
University of Illinois at Urbana-Champaign
1409 W. Green Street
Urbana IL 61801-2975
United States