# Kevin James. p-groups, Nilpotent groups and Solvable groups

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1 p-groups, Nilpotent groups and Solvable groups

2 Definition A maximal subgroup of a group G is a proper subgroup M G such that there are no subgroups H with M < H < G.

3 Definition A maximal subgroup of a group G is a proper subgroup M G such that there are no subgroups H with M < H < G. Theorem (p-groups) Let p be a prime and let P be agroup of order p a with a 1. Then, 1 Z(P) 1. 2 If 1 H P then H Z(P) 1. In particular, every normal subgroup of order p is contained in the center. 3 If H P, then for each p b H, K H such that K P. In particular, P has a normal subgroup of order p b for 1 b a. 4 If H < P then H < N P (H). 5 Every maximal subgroup of P is of index p and is normal in P.

4 Definition 1 For G a group we inductively define: Z 0 (G) = 1, Z 1 (G) = Z(G) and Z i+1 (G) is the subgroup of G containing Z i (G) such that Z i+1 (G)/Z i (G) = Z (G/Z i (G)) The chain Z 0 (G) Z 1 (G) Z 2 (G)... is called the upper central series of G. 2 A group G is called nilpotent if Z c (G) = G for some c Z. The smallest such c is called the nilpotence class of G.

5 Note 1 One can show (for homework) that Z i (G) char G, i. 2 If 1 G is Abelian then G is nilpotent of class 1. 3 For finite groups, the upper central series must stabilize. 4 If two terms of the upper central series are the same then the series stabilize form that point onward.

6 Note 1 One can show (for homework) that Z i (G) char G, i. 2 If 1 G is Abelian then G is nilpotent of class 1. 3 For finite groups, the upper central series must stabilize. 4 If two terms of the upper central series are the same then the series stabilize form that point onward. Proposition Let p be aprime and let P be a group of order p a. Then P is nilpotent of nilpotence class at most (a 1).

7 Theorem Let G be a finite group and let p 1, p 2,..., p s be the distince primes dividing G and let P i Syl pi (G), 1 i s. Then the following are equivalent 1 G is nilpotent. 2 If H < G then H < N G (H). 3 P i G for 1 i s. 4 G = P 1 P 2 P s.

8 Theorem Let G be a finite group and let p 1, p 2,..., p s be the distince primes dividing G and let P i Syl pi (G), 1 i s. Then the following are equivalent 1 G is nilpotent. 2 If H < G then H < N G (H). 3 P i G for 1 i s. 4 G = P 1 P 2 P s. Corollary A finite Abelian group is the direct product of its Sylow subgroups.

9 Proposition If G is a finite group such that for all positive integers n dividing its order, G contains at most n elements x satisfying x n = 1, then G is cyclic.

10 Proposition If G is a finite group such that for all positive integers n dividing its order, G contains at most n elements x satisfying x n = 1, then G is cyclic. Proposition (Frattini s Argument) Let G be a finite group, let H be a normal subgroup of G and let P be a Sylow p-subgroup of H. Then G = HN G (P) and [G : H] divides N G (P).

11 Proposition If G is a finite group such that for all positive integers n dividing its order, G contains at most n elements x satisfying x n = 1, then G is cyclic. Proposition (Frattini s Argument) Let G be a finite group, let H be a normal subgroup of G and let P be a Sylow p-subgroup of H. Then G = HN G (P) and [G : H] divides N G (P). Proposition A finite group is nilpotent if and only if every maximal subgroup is normal.

12 Definition For any group G we inductively define: G 0 = G, G 1 = [G, G], and G i+1 = [G, G i ]. Then chain of groups G 0 G 1... is called the lower central series of G.

13 Definition For any group G we inductively define: G 0 = G, G 1 = [G, G], and G i+1 = [G, G i ]. Then chain of groups G 0 G 1... is called the lower central series of G. Theorem A group G is nilpotent if and only if G n = 1 for some n 0. More precisely, G is nilpotent of class c if and only if c is the smallest nonnegative integer such that G c = 1. If G is nilpotent of class c then Z i (G) G c i 1 Z i+1 (G) 0 i c 1.

14 Recall A group G is said to be solvable if there is a series 1 = H 0 H 1 H s = G such that H i+1 /H i is Abelian.

15 Recall A group G is said to be solvable if there is a series 1 = H 0 H 1 H s = G such that H i+1 /H i is Abelian. Definition For any group G we inductively define: G (0 ) = G, G (1) = [G, G], and G (i+1) = [G (i), G (i) ] for all i 1. This series of groups is called the derived or commutator series of G.

16 Recall A group G is said to be solvable if there is a series 1 = H 0 H 1 H s = G such that H i+1 /H i is Abelian. Definition For any group G we inductively define: G (0 ) = G, G (1) = [G, G], and G (i+1) = [G (i), G (i) ] for all i 1. This series of groups is called the derived or commutator series of G. Theorem A group G is solvable if and only if G (n) = 1 for some n 0.

17 Definition It G is solvable, the smallest nonnetative n for which G (n) = 1 is called the solvable length of G.

18 Definition It G is solvable, the smallest nonnetative n for which G (n) = 1 is called the solvable length of G. Proposition Let G and K be groups, let H G and let φ : G K be a surjective homomorphism. 1 H (i) G (i), i 0. In particular, if G is solvable then so is H and the solvable length of H is less than or equal to the solvable length of G. 2 φ(g (i) ) = K (i). In particular, homomorphic images of quotient groups of solvable groups are solvable of solvable length less than or equal to that of the domain group. 3 If N G and both N and G/N are solvable then so is G.

19 Theorem Burnside If G = p a q b for some primes p and q, then G is solvable. Philip Hall If for every prime p dividing G, we write G = p a m with (p, m) = 1, and G has a subgroup of order m, then G is solvable. Feit-Thompson If G is odd then G is solvable. Thompson If for every pair of elements x, y G, < x, y > is solvable then so is G.

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