Finite groups with a special 2-generator property, and order of centralizers in finite groups

This paper deals with finite groups, and has two parts. In part I J. L. Brenner and James Wielgold (I,3) defined a finite nonabelian group G as lying in $\Gamma\sb1\sp{(2)}$ (spread one-two) if for every 1 $\not=$ x $\in$ G, either x is an involution and G = $\langle$x,y$\rangle$ for some y $\in$ G or x is not an involution and there is an involution z $\in$ G with G = $\langle$x,z$\rangle$. We show that "most" of the simple groups of Lie type do not lie in $\Gamma\sb1\sp{(2)}$, we classify all those solvable groups which lie in $\Gamma\sb1\sp{(2)}$, and we show that a finite non-simple non-solvable group lies in $\Gamma\sb1\sp{(2)}$ if it is isomorphic to the semi-direct product of N and $\langle$x$\rangle$ where x is an involution and N is a simple nonabelian group. Many simple groups are excluded from being candidates for the N above.Part II includes a characterization of all groups G having a subgroup A with $\vert$A$\Vert$C$\sb{\rm G}$(A)$\vert$ $>$ $\vert$G$\vert$, and those for which m$\sb1$ = sup $\{\vert$B$\Vert$C$\sb{\rm G}$(B)$\vert$: B $\le$ G$\}$ = $\vert$G$\vert$. It is shown also that if G is not a direct product, then either there exists a nontrivial characteristic abelian subgroup A of G with $\vert$A$\Vert$C$\sb{\rm G}$(A)$\vert$ $\ge$ $\vert$G$\vert$, or $\vert$B$\Vert$C$\sb{\rm G}$(B)$\vert$ $<$ $\vert$G$\vert$ for any proper nontrivial subgroup B of G.U of I OnlyETDs are only available to UIUC Users without author permissio