Talk:M-group

Old definition

In the original version of this article, a group was defined to be an M-group if "G/C_G(F(G)) is nilpotent". I have found no sources backing up this claim, and many sources backing up the two definitions provided now. I deleted the original definition as unsourced. To demonstrate it is not just equivalent to a standard definition, here are some remarks. I will call a group a VM-group if G/C_G(F(G)) is nilpotent.

Firstly "VM" and "monomial" are not equivalent, and only mildly related:

• There are monomial groups that are not VM: The symmetric group of order 24 is a monomial group, but the centralizer of its Fitting subgroup is K4 with quotient isomorphic to S3, so not nilpotent.
• There are VM-groups that are not monomial: The simple group A5 of order 60 is VM, but every monomial group is sovlable by Taketa's theorem (Isaacs 5.13).
• Every solvable VM-group is a monomial group (Isaacs 6.22 and 6.23 show that if N is an A-group and G/N is nilpotent, then G is a monomial group, and in a solvable group N=C_G(F(G)) is abelian).

Secondly, "VM" and "modular" are not equivalent:

• There are VM groups that are not modular: The dihedral group of order 8 and the nonabelian group of exponent p and order p^3 are both VM but not modular. Since every nilpotent group is VM, this probably means the ideas are unrelated.
• There are modular groups that are not VM: extended Tarski monsters are modular, but not VM
• I believe every finite modular group is VM by Iwasawa's structure theorem.

At any rate, the VM definition appears to be unsourceable, but I left it here in case someone wants to give it a try. JackSchmidt (talk) 23:38, 12 February 2008 (UTC)