Vol. 44, No. 2, 1973

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Discontinuous characters and subgroups of finite index

Harold L. Peterson, Jr.

Vol. 44 (1973), No. 2, 683–691

For brevity’s sake, a subgroup of finite index is called “large.” A discontinuous torsion character is (clearly) one whose kernel is a large nonopen subgroup. Compact Abelian groups (and certain other LCA groups) have the same number (either none or infinitely many) of large nonopen subgroups and discontinuous torsion characters. Locally compact Abelian groups of which all large subgroups are open include connected, locally connected, and monothetic (possibly totally disconnected) groups. Contrariwise, there are locally compact groups G which have as many as 2|𝜃| large nonopen subgroups. These include nondiscrete torsion Abelian groups of bounded order and all totally disconnected, nonmetrizable, compact groups.

Mathematical Subject Classification 2000
Primary: 22D05
Received: 4 October 1971
Revised: 30 August 1972
Published: 1 February 1973
Harold L. Peterson, Jr.