Page 142 - DMTH403_ABSTRACT_ALGEBRA

Unit 13: Solvable Groups

The smallest subgroup that contains all commutators of G is called the commutator subgroup or Notes
derived subgroup of G, and is denoted by G’.
Proposition: 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: Let G be a group. The subgroup (G’ )’ is called the second derived subgroup of G. We
define G inductively as (G (k-1) )’, and call it the k th derived subgroup.
(k)
Theorem 2: A group G is solvable if and only if G = {e} for some positive integer n.
(n)
Corollary: 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: Let G be a group. A chain of subgroups G = N N ... N such that
0
1
n
(i) N is a normal subgroup in N for i = 1, 2, . . . ,n,
i
i-1
(ii) N / N is simple for i = 1, 2, . . . ,n, and
i-1
i
(iii) N = {e}
n
is called a composition series for G.
The factor groups N / N are called the composition factors determined by the series.
i-1
i
Theorem 3: [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.
Example: Let p be a prime and let G be a non-abelian group of order p . Show that the
3
center Z(G) of G equals the commutator subgroup G’ of G.
Solution: Since G is non-abelian, we have |Z(G)| = p. (The center is nontrivial, and if |Z(G)| =
p , then G/Z(G) is cyclic, the text implies that G is abelian.) On the other hand, any group of
2
order p is abelian, so G/Z(G) is abelian, which implies that G’ Z(G). Since G is nonabelian, G’ {e},
2
and therefore G’ = Z(G).

Example: Prove that D is solvable for all n.
n
One approach is to compute the commutator subgroup of D , using the standard description
n
D = { a b | 0  i < n, 0  j < 2, o(a) = n, o(b) = 2, ba = a b }
-1
i
j
n
We must find all elements of the form xyx y , for x,y in D . We consider the cases x = a or x = a b
-1 -1
i
i
n
and y = a or y = ab.
j
j
Case 1: If x = a and y = a, the commutator is trivial.
i
j
Case 2: If x = a and y = ab, then xyx y = a aba ab = a aa bab = a aa a b = a , and thus each even
j
j
i
2i
i j i -j 2
i j
-i j
i j i
-1 -1
power of a is a commutator.
Case 3: If x = ab and y = a , we get the inverse of the element in Case 2.
i
j
Case 4: If x = a b and y = ab, then
j
i
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