James Beidleman
- Algebra and Number Theory
- group theory
My research interest is in Group Theory. Recently I have been concerned with products of groups, permutable subgroups, and finite groups in which the subnormal subgroups are normal or permute with certain classes of subgroups.
Let A and B be subgroups of a group G. A is said to permute with B if AB = { ab | a in A, b in B } is a subgroup of G. A is said to be permutable in G if A permutes with each subgroup of G. A and B is said to be a totally permutable (mutually permutable) pair provided that each subgroup of A permutes with each subgroup of B (A permutes with each subgroup of B and B permutes with each subgroup of A). A is said to be S-permutable if it permutes with all the Sylow subgroups of G.
Let the group G be a product of the subgroups A and B. A question that has been of interest for some time is: if A and B satisfy certain properties, what can be said about G? For example, if A and B are abelian (resp. nilpotent) and G is finite, then G is metabelian (resp. soluble). Another result is that if A and B form a totally permutable pair of supersoluble subgroups, then G is supersoluble.
A group G is called a T (resp. PT)-group provided that normality (resp. permutability) is transitive in G. That is, G is a T (resp. PT)-group provided that whenever H and K are subgroups of G such that H is normal (resp. permutable) in K and K is normal (resp. permutable) in G, then H is normal (resp. permutable) in G. A finite group is called a PST-group if S-permutability is transitive. A finite group is a PT (resp. PST)-group if all the subnormal subgroups of G are permutable (resp. S-permutable).
Recent publications on some of these topics include:
- J.C. Beidleman, A. Galoppo, H. Heineken and M. Manfredino, On certain products of soluble groups, Forum Mathematicum 13 (2001), 569-580.
- J.C. Beidleman and H. Heineken, Totally permutable torsion groups, J. Group Theory 2 (1999), 377-392.
- J.C. Beidleman, B. Brewster and D.J.S. Robinson, Criteria for permutability to be transitive in finite groups, J. Algebra 222 (1999), 400-412.
- J.C. Beidleman and H. Heineken, On the Hyperquasicenter of a Group, Journal of Group Theory 4 (2001), 199-206.
- J.C. Beidleman and H. Heineken, A Survey of Mutually and Totally Permutable Products in Infinite Groups, Quaderni di Matematica 8 (2002), 47-62.
- J.C. Beidleman and H. Heineken, Finite Soluble Groups whose Subnormal Subgroups Permute with Certain Classes of Subgroups, Journal of Groups Theory 6 (2003), 139-158.
- A. Ballester-Bolinches, J.C. Beidleman and H. Heineken, Groups in which Sylow Subgroups and Subnormal Subgroups Permute, Illinois Journal of Math. 47 (2003), 63-69.
- J.C. Beidleman and H. Heineken, Pronormal and Subnormal Subgroups and Permutability, Bollettino dell'Unione Mat. Italiana 6b (2003), 605-615.
- A. Ballester-Bolinches, J.C. Beidleman and H. Heineken, A Local Approach to Certain Classes of Finite Groups, Comm. in Algebra 31 (2003), 5931-5942.
- J.C. Beidleman, P. Hauck and H. Heineken, Totally Permutable Products of Certain Classes of Finite Groups, J. Algebra 276 (2004), 826-835.
- J.C. Beidleman and H. Heineken, Pairwise N-connected Products of Certain Classes of Finite Groups, Comm. in Algebra 32 (2004), 4741-4752.
- J.C. Beidleman and H. Heineken, Mutually Permutable Subgroups and Group Classes, Archiv der Mathematik 85 (2005), 18-30.