Abelian-to-normal replacement fails for prime-sixth order for prime equal to two
History
This result appeared in a paper by Alperin in 1965.
Statement
For , it is possible to have a finite -group that has an abelian subgroup of order but no abelian normal subgroup of that order.
The smallest example is the case where the order of the group is .
Related facts
- Glauberman's abelian-to-normal replacement theorem for bounded exponent and half of prime plus one: This states that if , the existence of an abelian subgroup of order implies the existence of an abelian normal subgroup of order . In particular, for , it shows that the existence of an abelian subgroup of order implies the existence of an abelian normal subgroup of order .
- Abelian-to-normal replacement theorem for prime-cube order
- Abelian-to-normal replacement theorem for prime-fourth order
- Congruence condition on number of abelian subgroups of prime-cube order
- Congruence condition on number of abelian subgroups of prime-fourth order
- Abelian-to-normal replacement fails for prime-cube index for prime equal to two: The same example works.
- Abelian-to-normal replacement theorem for prime-cube index for odd prime: This was proved by Alperin and later in a paper by Jonah and Konvisser.
Proof
Further information: Free product of class two of two Klein four-groups
Let be the quotient of the free product of two copies of the Klein four-group by the third member of its lower central series. In other words, is the free product of class two of two copies of the Klein four-group.
Let be the semidirect product of by an automorphism of order two that interchanges the two copies. Then, is a group of order . We claim that has exactly two abelian subgroups of order , both are elementary abelian, and neither is normal in .
References
Journal references
- Large abelian subgroups of p-groups by Jonathan Lazare Alperin, Transactions of the American Mathematical Society, Volume 117, Page 10 - 20(Year 1965): ^{Official copy}^{More info}, Page 11, shortly after Theorem 4.