| Summary: | <p>Using combinatorial and character-theoretic methods the following theorems are proved.</p> <p><em>Theorem I</em> Let G be a finite simple group containing two non-conjugate subgroups A, B such that:</p> <p><ol type="i"><li><p>if a ε A, a &notequals; 1, then C<sub>G</sub> (a) = A; if b ε B, b &notequals; 1, then</p> <p>C<sub>G</sub>(b) = B;</p></li> <li>| N<sub>G</sub>(A)/A | = 2, | N<sub>G</sub>(B)/B | = 2.</li></ol></p> <p>Then G is isomorphic to some PSL (2,2<sup>a</sup>).</p> <p><em>Theorem II</em> Let G be a finite simple group containing a subgroup A such that:</p> <p><ol type="i"><li>if a ε A, a &notequals; 1, then C<sub>G</sub>(a) = A;</li> <li>| N<sub>G</sub>(A)/A | = 2;</li> <li>3 divides | A | .</li></ol></p> <p>Then G is isomorphic to some PSL (2,q).</p>
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