Since L S n A n and L A n A n E i A n for i 1 There is no Fitting length

Since l s n a n and l a n a n e i a n for i 1 there

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Since L ( S n ) = A n and L ( A n ) = A n , E i = A n for i 1. There is no Fitting length. Theorem 3.8. A finite group has a Fitting length if and only if it is solvable. Proof. Suppose G is a finite solvable group. If F i 6 = G then G/F i is a nontrivial finite solvable group so its Fitting subgroup F i +1 /F i is nontrivial. Therefore F i +1 6 = F i , so for large i we must have F i = G . Similarly, if E i 6 = { e } then E i +1 6 = E i (because E i +1 =
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SUBGROUP SERIES II 7 L ( E i ) L 1 ( E i ) = E 0 i , which is a proper subgroup of E i because E i is a nontrivial solvable group). Therefore E i is trivial for large i . Now suppose G has a Fitting length. The upper nilpotent series is a normal series for G with nilpotent factors. Since nilpotent groups are solvable and solvability of N and H/N implies solvability of H , we get solvability of G by arguing inductively that every F i is solvable. Remark 3.9. A group G is called metacyclic , metabelian , or metanilpotent if it has a normal subgroup N such that N and G/N are both cyclic, both abelian, or both nilpotent. This means the normal series { e } C N C G has both factors cyclic or abelian or nilpotent, so these properties are preserved by passage to subgroups and quotient groups (but not direct products, since the direct product need not have such a series with 2 factors). All such groups are solvable since N and G/N are solvable. Check that metabelian is the same as having solvable length 2 while (when G is finite) metanilpotent is the same as having Fitting length 2. In addition to the Fitting subgroup, there is another important nilpotent subgroup of any finite group: the Frattini subgroup. Definition 3.10. The Frattini subgroup of a finite group is the intersection of its maximal subgroups: Φ( G ) = \ max. M M. We set the trivial group to have trivial Frattini subgroup. The intersection defining Φ( G ) is preserved by conjugations, so Φ( G ) C G . For instance, Φ( D 4 ) = { 1 , r 2 } and Φ( S n ) is trivial for n 3. (In particular, since D 4 is the 2-Sylow subgroup of S 4 , we see that if H G then Φ( H ) can be larger than Φ( G ).) When G is nilpotent, G 0 Φ( G ) by Theorem 2.2(2), so the quotient G/ Φ( G ) is abelian. Much more can be said (and used) about this quotient when G is a finite p -group: look up the “Burnside basis theorem” in a group theory book. Theorem 3.11 (Frattini, 1885) . For any finite group G , Φ( G ) is nilpotent. Proof. We will show all the Sylow subgroups of Φ( G ) are normal subgroups of Φ( G ). Let P be a Sylow subgroup of Φ( G ). Then G = Φ( G ) N G ( P ) (Frattini argument; see the handout on applications of the Sylow theorems). If P is not normal in G then N G ( P ) 6 = G . Let M be a maximal subgroup of G containing N G ( P ), so Φ( G ) M . Therefore G = Φ( G ) N G ( P ) M , a contradiction, so P C G , which implies P C Φ( G ). Since, for any finite group G , Φ( G ) is a nilpotent normal subgroup of G , Φ( G ) F( G ) .
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