Page 145 - DMTH403_ABSTRACT_ALGEBRA

Abstract Algebra

Notes 13.3 Keywords

Commutator Subgroup: The smallest subgroup that contains all commutators of G is called the
commutator subgroup or derived subgroup of G, and is denoted by G’.
Let G be a group. A chain of subgroups G = N N ... N such that
1
0
n
(i) N is a normal subgroup in N for i = 1, 2, . . ., n,
i-1
i
(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.
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
13.4 Review Questions

1. Prove the normal series
Z  { 3 }  { 15 }  { 0 }
60
Z  { 4 }  { 20 }  { 0 }
60
of the group Z are isomorphic.
60
2. Let G and H be solvable groups. Show G × H is also solvable.
3. If G has a composition series and if N is a proper normal subgroup of G, Show the n exists
a composition series containing N.

4. Let N be a normal subgroup of G. If N and G/N have composition series, then G must also
have a composition series.
5. Let N be a normal subgroup of G if N and G/N are solvable groups. Show that G is also
solvable group.
6. Prove that G is a solvable group if and only if G has a series of subgroups G = P  P n-1
n
 ... P  P = { e }
0
1
where p is normal in p and the order p /p is prime.
i
i
i+1
i+1
Answers: Self Assessment
1. (c) 2. (a) 3. (a) 4. (b)
13.5 Further Readings

Books Dan Saracino: Abstract Algebra; A First Course.
Mitchell and Mitchell: An Introduction to Abstract Algebra.
John B. Fraleigh: An Introduction to Abstract Algebra (Relevant Portion).

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