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In mathematical finite group theory, the Thompson subgroup J(P) of a finite p-group P refers to one of several characteristic subgroups of P. Thompson (1964) originally defined J(P) to be the subgroup generated by the abelian subgroups of P of maximal rank. More often the Thompson subgroup J(P) is defined to be the subgroup generated by the abelian subgroups of P of maximal order or the subgroup generated by the elementary abelian subgroups of P of maximal rank. In general these three subgroups can be different, though they are all called the Thompson subgroup and denoted by J(P).
See also

Glauberman normal p-complement theorem
ZJ theorem
Puig subgroup, a subgroup analogous to the Thompson subgroup

References

Gorenstein, D.; Lyons, Richard; Solomon, Ronald (1996), The classification of the finite simple groups. Number 2. Part I. Chapter G, Mathematical Surveys and Monographs 40, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-0390-5, MR 1358135
Thompson, John G. (1964), "Normal p-complements for finite groups", Journal of Algebra 1: 43–46, doi:10.1016/0021-8693(64)90006-7, ISSN 0021-8693, MR 0167521
Thompson, John G. (1969), "A replacement theorem for p-groups and a conjecture", Journal of Algebra 13: 149–151, doi:10.1016/0021-8693(69)90068-4, ISSN 0021-8693, MR 0245683

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