On -groups with no normal abelian subgroups of rank , and their occurrence as Sylow -subgroups of finite simple groups

Author:
Anne R. MacWilliams

Journal:
Trans. Amer. Math. Soc. **150** (1970), 345-408

MSC:
Primary 20.29

DOI:
https://doi.org/10.1090/S0002-9947-1970-0276324-3

MathSciNet review:
0276324

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We prove that in a finite -group with no normal Abelian subgroup of rank , every subgroup can be generated by four elements. This result is then used to determine which -groups with no normal Abelian subgroup of rank can occur as 's of finite simple groups , under certain assumptions on the embedding of in .

**[1]**J. Alperin,*Centralizers of Abelian normal subgroups of -groups*, J. Algebra**1**(1964), 110-113. MR**29**#4800. MR**0167528 (29:4800)****[2]**N. Blackburn,*On a special class of -groups*, Acta Math.**100**(1958), 45-92. MR**21**#1349.**[3]**-,*Generalizations of certain elementary theorems on -groups*, Proc. London Math. Soc. (3)**11**(1961), 1-22. MR**23**#A208.**[4]**R. Brauer,*Some applications of the theory of blocks of characters of finite groups*. II, J. Algebra**1**(1964), 307-334. MR**30**#4836.**[5]**W. Feit,*Characters of finite groups*, Notes printed by Yale University, New Haven, Conn., 1965.**[6]**W. Feit and J. Thompson,*Solvability of groups of odd order*, Pacific J. Math.**13**(1963), 775-1029. MR**29**#3538.**[7]**G. Glauberman,*Central elements in core-free groups*, J. Algebra**4**(1966), 403-420. MR**34**#2681.**[8]**G. Higman,*Suzuki -groups*, Illinois J. Math.**7**(1963), 79-96. MR**26**#1365.**[9]**B. Huppert,*Endliche Gruppen*. I, Die Grundlehren der math. Wissenschaften, Band 134, Springer-Verlag, Berlin and New York, 1967. MR**37**#302.**[10]**J. Thompson,*Non-solvable finite groups whose nonidentity solvable subgroups have solvable normalizers*(to appear).

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
20.29

Retrieve articles in all journals with MSC: 20.29

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1970-0276324-3

Keywords:
-group,
Sylow -subgroup,
automorphism,
rank,
generators,
conjugate,
transfer homomorphism,
fusion

Article copyright:
© Copyright 1970
American Mathematical Society