Brauer-Fowler theorem on existence of subgroup of order greater than the cube root of the group order
From Groupprops
Contents
Statement
Suppose is a finite group whose order is an even number greater than 2. Then, has a proper subgroup such that:
Further, if the center is an odd-order group, then we can choose to be the centralizer of some non-identity strongly real element of .
Related facts
Similar facts
- Dihedral trick
- Finite group having at least two conjugacy classes of involutions has order less than the cube of the maximum of orders of centralizers of involutions
- 2-Sylow subgroup is TI implies it is normal or there is exactly one conjugacy class of involutions
Opposite facts
- Finite simple non-abelian group has order greater than product of order of proper subgroup and its centralizer
- Every proper abelian subgroup of a finite simple non-abelian group has order less than its square root
References
Textbook references
- Finite Groups by Daniel Gorenstein, ISBN 0821843427, Page 303, Chapter 9 (Groups of even order), Theorem 1.6, ^{More info}