Solvable generation of groups and Sylow subgroups of the lower central series (original) (raw)

The classification of finite simple groups has many conseqttenccs, one of which is THEOREM A. Let G be a finite group. There exist a sokable subgroup 5 and g E G such that G = (S, SK). 'This theorem is proved in Section 2. An example is given to show "solvable" cannot be replaced by "nilpotent." Theorem A allows one to extend results on solvable groups. In Section 3, a particular theorem about solvable groups is obtained. THEOREM B. Suppose #:G-+ GL(V) is a nontrivial irreducible representation (otler anql field) of a finite solvable group G. If M and N are maxim.al subgroups of G such that C,(M) # 0 # C,(N), then M and A' are conjugate. Theorem B fails to hold in general. However, it does verify the following conjecture in the solvable case. :* Supported in part by the National Science Foundation.