Excerpted from Beachy/Blair, Abstract Algebra, 2nd Ed. © 1996

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 · · · Nn such that

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

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

(iii) Nn = {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 · · · Nn such that

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

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

(iii) Nn = {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 . . . Nn such that

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

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

(iii) Nn = {e}
is called a composition series for G.
    The factor groups Ni-1 / Ni are called the composition factors determined by the series.

 
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 Dn 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