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Rotman, Joseph J. Foguel, Tuval Shmuel 2011-05-07T12:30:43Z 2011-05-07T12:30:43Z 10000-01-01 1992 "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. Made available in DSpace on 2011-05-07T12:30:43Z (GMT). No. of bitstreams: 2 license.txt: 4922 bytes, checksum: 910b249b4beec47e7ab768910c8f966f (MD5) 9236461.pdf: 1841647 bytes, checksum: 0b79a8c8d1588eea079c69288eb5e206 (MD5) Previous issue date: 1992 Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Howard Ding (hding2@illinois.edu) on 2011-05-07T14:41:58Z Item is restricted indefinitely. Restriction data tranferred 2014-07-01T11:18:13-05:00 Original Data Group with Access UIUC Users [automated] Release Date: none Reason: ETDs are only available to UIUC Users without author permission ETDs are only available to UIUC Users without author permission U of I Only AAI9236461 (UMI)AAI9236461 http://hdl.handle.net/2142/20160 eng Copyright 1992 Foguel, Tuval Shmuel Mathematics Finite groups with a special 2-generator property, and order of centralizers in finite groups text Mathematics Mathematics University of Illinois at Urbana-Champaign Dissertation Ph.D.