Forward to §7.7 | Back to §7.5 | Up | Table of Contents | About this document

§ 7.6 Solvable Groups

 
Definition 7.6.1. The group G is said to be solvable if there exists a finite chain of subgroups
G = N₀ N₁ · · · N_n such that

(i) N_i is a normal subgroup in N_i-1 for i = 1,2, . . . ,n,

(ii) N_i-1 / N_i is abelian for i = 1,2, . . . ,n, and

(iii) N_n = {e}.

 
Proposition 7.6.2. A finite group G is solvable if and only if there exists a finite chain of subgroups
G = N₀ N₁ · · · N_n such that

(i) N_i is a normal subgroup in N_i-1 for i = 1,2, . . . ,n,

(ii) N_i-1 / N_i is cyclic of prime order for i = 1,2, . . . ,n, and

(iii) N_n = {e}.

 
Theorem 7.6.3. Let p be a prime number. Any finite p-group is solvable.

 
Definition 7.6.4. Let G be a group. An element g in G is called a commutator if

g = aba^-1b^-1

for elements a,b in G.
    The smallest subgroup that contains all commutators of G is called the commutator subgroup or derived subgroup of G, and is denoted by G'.

 
Proposition 7.6.5. Let G be a group with commutator subgroup G'.

(a) The subgroup G' is normal in G, and the factor group G/G' is abelian.

(b) If N is any normal subgroup of G, then the factor group G/N is abelian if and only if G' N.

 
Definition 7.6.6. Let G be a group. The subgroup (G' )' is called the second derived subgroup of G. We define G^(k) inductively as (G^(k-1))', and call it the k th derived subgroup

 
Theorem 7.6.7. A group G is solvable if and only if G⁽ⁿ⁾ = {e} for some positive integer n.

 
Corollary 7.6.8. Let G be a group.

(a) If G is solvable, then so is any subgroup or homomorphic image of G.

(b) If N is a normal subgroup of G such that both N and G/N are solvable, then G is solvable.

 
Definition 7.6.9. Let G be a group. A chain of subgroups
G = N₀ N₁ . . . N_n such that

(i) N_i is a normal subgroup in N_i-1 for i = 1,2, . . . ,n,

(ii) N_i-1 / N_i is simple for i = 1,2, . . . ,n, and

(iii) N_n = {e}

 
Theorem 7.6.10. [Jordan-Hölder] Any two composition series for a finite group have the same length. Furthermore, there exists a one-to-one correspondence between composition factors of the two composition series under which corresponding composition factors are isomorphic.

§ 7.6 Solvable Groups: Solved problems

 
9. Let p be a prime and let G be a nonabelian group of order p³. Show that the center Z(G) of G equals the commutator subgroup G' of G.     Solution

10. Prove that D_n is solvable for all n.     Solution

11. Prove that any group of order 588 is solvable, given that any group of order 12 is solvable.     Solution

12. Let G be a group of order 780=2² · 3 · 5 · 13. Assume that G is not solvable. What are the composition factors of G? (Assume that the only nonabelian simple group of order 60 is A₅.)     Solution

Printable solutions to the problems | Forward to §7.7 | Back to §7.5 | Up | Table of Contents