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

**Definition 7.6.1.**
The group G is said to be
** solvable**
if there exists a finite chain of subgroups

G =
N_{0}
N_{1}
** · · · **
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 = N_{0}
N_{1}
** · · · **
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^{-1}b^{-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 =
N_{0}
N_{1}
. . .
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}

The factor groups N

**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.

**9.**
Let p be a prime and let G be a nonabelian group of order p^{3}.
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=2^{2} **·** 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 A_{5}.)
*Solution*

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