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I had asked that if $G$ is a direct product of a $2$-group and a simple group, then is it possible that $G$ be a solvable group. That the answer is no! But by a Remark on T.M. Gagen, Topics in Finite Groups, London Math. Soc. Lecture Note Ser., vol. 16, Cambridge Univ. Press, Cambridge, 1976. I am confused! I will write the Remark below:

Definition 11.3. A group $G$ with an abelian Sylow $2$-subgroup is said to be an $A^*$-group, if $G$ has a normal series $1\subseteq N \subseteq M \subseteq G$ where $N$ and $G/M$ are of odd order and $M/N$ is a direct product of a $2$-group and simple groups of type $L_2(q)$ or $JR$.

Theorem A. Let $G$ be a finite group with an abelian Sylow $2$-subgroup. Then $G$ is an $A^*$-group.

Remark. If a group $G$ has an abelian Sylow $2$-subgroup $T$ of rank $1$, then $G$ is solvable and $2$-nilpotent and clearly an $A^*$-group. If $T$ has rank $2$, then $G$ is $2$-nilpotent unless $T$ is of type $( 2^\alpha , 2^\alpha )$ . But then if $\alpha > 1$, $G$ is solvable by [7] and clearly an $A^*$-group since it has $2$-length $1$. Thus we may assume that $|T | = 4$ and then we can apply the results of [13]. Again $G$ is an $A^*$-group.

By the Remark, it is possible that $G$ (with abelian sylow $2$-group) be a solvable group. In this case $M$ and then $M/N$ must be solvable, and can not be have a simple group as its direct factor. a contradiction with Definition!

Am I right?!

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  • $\begingroup$ I think you are. Either some confusion exists there with some definition (do you have a direct link to that paper?) or else that could be a rather major blunder...Unless the remark should have remarked the rank 1 thing, which perhaps is what makes $\,G\,$ solvable! $\endgroup$ – DonAntonio Jun 25 '13 at 18:57
  • $\begingroup$ Please see here for how to use Markdown formatting (italics is not the same thing as math mode). $\endgroup$ – Zev Chonoles Jun 25 '13 at 18:58
  • $\begingroup$ Hmmm...yet there still is a problem, me believes... $\endgroup$ – DonAntonio Jun 25 '13 at 19:03
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This is a matter of convention. When $G$ is solvable, $M/N$ is an (abelian) 2-group. It is still the case that $M/N$ is the direct product of a 2-group and a set of simple groups, each isomorphic to $L_2(q)$ or $J_1$ or $^2G_2(q)$. In case $G$ is solvable, that set of simple groups is empty.

The remark is trying to clarify what happens in the low rank cases, where typically the result is a solvable group. In the $2^a$ case, Cayley showed you can take $G/M=1$, $M/N$ the cyclic Sylow 2-subgroup. In the $2^a \times 2^a$ ($a>1$) case Brauer's result shows $G/M$ can be taken to have order 1 or 3, and $M/N$ to be the homocyclic Sylow 2-subgroup. In the non-homocyclic case, Burnside or Frobenius shows you can take $G/M=1$ and $M/N$ to be the Sylow 2-subgroup. Only in the case $C_2 \times C_2$ (amongst rank 1 or 2) do you get to the case supporting a non-solvable $M/N$.

Rank 3 is interesting because of $J_1$ and $^2G_2(q)$.

I'll mention that Gagen, Bender, and Gorenstein all use the same phrasing, and all intend to allow the groups to be solvable as well as non-solvable.

Another small point: Gagen and Bender use the name “$A^*$-group” rather than “$A$-group” used by Gorenstein. The definitions are actually nearly identical; I believe they are trying to avoid confusion with a similar (weaker) result classifying A-groups in the sense of Hall: groups in which all Sylow subgroups are abelian.

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  • $\begingroup$ Thanks a lot. Is there any thing(book, paper, ...) about solvable groups. I need some property of solvable groups for my thesis, but I can't find what I need!!! I have read Rose and Gorenstein. Thanks again. $\endgroup$ – Adeleh Jun 25 '13 at 20:02
  • $\begingroup$ Doerk–Hawk Finite Solvable Groups and Huppert Endliche Gruppen are pretty thorough for solvable groups. What sort of (already known) results are you looking for? $\endgroup$ – Jack Schmidt Jun 25 '13 at 20:27
  • $\begingroup$ Thanks for your suggestion. I need some things about property of sylow subgroups of a solvable group and some thing else that may be very simple or may not be true!! For example is it always true that $Z(G)\leq \Phi(G)$ where $G$ is solvable? and other things like this. I hope that the first book be helpful. The second is germany, and I can't use it however I think that is very useful book. $\endgroup$ – Adeleh Jun 26 '13 at 7:34
  • $\begingroup$ Generally “central” and “Frattini” have a similar feel, but they are independent. For instance the center of the cyclic group of order 2 is large, while the frattini is small, but the dihedral group of order 18 has a large frattini subgroup and a small center. The central chief factors are covered by the system normalizers, and the frattini chief factors are covered by the pre-frattini subgroups. The largest normal subgroup contained in a system normalizer is the hypercenter (formed by the longest chain of centrals); in a prefrattini is the frattini (formed by the longest chain of frattinis). $\endgroup$ – Jack Schmidt Jun 26 '13 at 14:16

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