document.write( "Question 1131289: Prove that no group of order 96 is simple. \n" ); document.write( "

Algebra.Com's Answer #747997 by MathLover1(20850) $\"\"$ $\"About$

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\n" ); document.write( " $\"96=2%5E5%2A3\"$ \r
\n" ); document.write( "\n" ); document.write( "Theorem: Suppose that G is a finite group and p a prime such that np(G)>1, and choose distinct Sylow p-subgroups S and T of G such that the order |S∩T| is as large as possible. Then np(G)≡1(mod|S:S∩T|)\r
\n" ); document.write( "\n" ); document.write( "Applying this for |G|=96 and p=2 (assuming n[2](G)=3, otherwise we are done), we see that
\n" ); document.write( "|S:S∩T|=2.
\n" ); document.write( "Let D=S∩T, so D is normal in S and T.
\n" ); document.write( "Hence both S and T lie in the normalizer NG(D).
\n" ); document.write( "But S and T are distinct, hence |NG(D)|>|S|=25.
\n" ); document.write( "This forces NG(D)=G, i.e. D⊲G. We found a nontrivial proper normal subgroup.\r
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\n" ); document.write( "\n" ); document.write( "or\r
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\n" ); document.write( "\n" ); document.write( "There is a following generalization of one of the Sylow's theorems (see for example here), which is equivalent to the theorem in Cihan's answer.\r
\n" ); document.write( "\n" ); document.write( "Let $\"P%5B1%5D\"$ , $\"P%5B2%5D\"$ ,…, $\"P%5Bk+%5D\"$ be the Sylow p-subgroups of G. If [ $\"P%5Bi%5D\"$ : $\"P%5Bi%5D\"$ ∩ $\"P%5Bj%5D\"$ ]≥ $\"p%5Ed\"$ whenever i≠j, then $\"n%5Bp%5D\"$ ≡1mod $\"p%5Ed\"$ \r
\n" ); document.write( "\n" ); document.write( ".Taking $\"d=1\"$ gives you one of Sylow's theorems, and the contrapositive of this theorem in the case $\"d=2\"$ is useful here.
\n" ); document.write( "If $\"n%5Bp%5D\"$ ≢1mod $\"p%5E2\"$ , then there exist different Sylow p-subgroups $\"P\"$ and $\"+Q\"$ with [ $\"P\"$ : $\"P\"$ ∩ $\"Q\"$ ]< $\"p%5E2\"$ .
\n" ); document.write( "This implies that [ $\"P\"$ : $\"P\"$ ∩ $\"Q\"$ ]=[ $\"Q\"$ : $\"P\"$ ∩ $\"Q\"$ ]= $\"p\"$ , and thus the intersection $\"P\"$ ∩ $\"Q\"$ is normal in both $\"P\"$ and $\"Q\"$ .\r
\n" ); document.write( "\n" ); document.write( "In your problem, we can assume $\"n%5B2%5D=3\"$ . Then $\"3\"$ ≢1mod $\"4\"$ , so we find different Sylow 2-subgroups $\"P\"$ and $\"Q\"$ with $\"+P\"$ ∩ $\"Q\"$ normal in both $\"P\"$ and $\"Q\"$ .
\n" ); document.write( "Therefore the intersection $\"P\"$ ∩ $\"Q\"$ is normal in the subgroup ⟨ $\"P\"$ , $\"Q\"$ ⟩ generated by $\"P\"$ and $\"Q\"$ . The order of ⟨ $\"P\"$ , $\"Q\"$ ⟩ is a multiple of $\"2%5E5\"$ and larger than $\"2%5E5\"$ , so it has to equal $\"G\"$ .
\n" ); document.write( "Thus $\"P\"$ ∩ $\"Q\"$ is a nontrivial proper normal subgroup of $\"+G\"$ .\r
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