Chains of Normalizers of Subnormal Subgroups

Abstract

We investigate properties of a few series associated to subnormal subgroups of finite groups. For a subnormal subgroup of a finite group we show that there are subnormal series for which the normalizers of each subgroup in the series form a chain. This allows us to formulate a bound on the subnormal depth of a subgroup using the index of its normalizers in the original group.

MSC::

• Primary 20E15
• Secondary 20F22
• 20D35

Acknowledgment

The first author was partially supported by a General Omar N. Bradley Research Fellowship in Mathematics. The views expressed in this article are those of the authors and do not necessarily reflect those of ARCYBER, the U.S. Army, or the Department of Defense.

Additional information

Notes on contributors

William Cocke

William Cocke is serving as an Officer in the U.S. Army. He received his Ph.D. in Mathematics from the University of Wisconsin and also has a B.S. and an M.S. from Brigham Young University. Currently, he is also pursuing a Ph.D. in Computer and Cyber Sciences from Augusta University. Along with his wife and children he runs a small goat farm.

I. M. Isaacs

I. M. Isaacs attended the famous Bronx High School of Science, and then the Polytechnic Institute of Brooklyn. He received a Ph.D. from Harvard University with a dissertation in group representation theory, under the supervision of Richard Brauer. After his postdoctoral position at the University of Chicago he took a professorship at the University of Wisconsin, Madison, where he remained until his retirement in 2012. He is very proud of the fact that he supervised about two dozen Ph.D. dissertations.

Ryan McCulloch

Ryan McCulloch is an associate professor of mathematics at Elmira College. He received his Ph.D. in 2014 from Binghamton University under the advisement of Ben Brewster. He has previously taught at University of Bridgeport and SUNY Oswego. He is a proud native of Pittston, Pennsylvania and enjoys growing Pittston tomatoes.