Page 144 - DMTH403_ABSTRACT_ALGEBRA

Unit 13: Solvable Groups

Self Assessment Notes

1. Let P be a prime number. Any ............... p-group is solvable
(a) infinite (b) direct

(c) finite (d) indirect
2. The smallest subgroup that contains all commutations of G is called as ...............
(a) commutator subgroup (b) normal subgroup

(c) generator subgroup (d) cyclic subgroup
3. If x = a and y = ..............., the commutator is trivial
i
(a) a j (b) a -1
(c) a -j (d) y -1
4. Let G be a group the subgroup is called the ............... of G.

(a) normal subgroup (b) second derived subgroup
(c) composition series (d) cyclic series

13.2 Summary

The group G is said to be solvable if there exists a finite chain of subgroups G = N N
 0 1
··· N such that
n
(i) N is a normal subgroup in N for i = 1, 2, ..., n,
i
i-1
(ii) N / N is abelian for i = 1, 2, ..., n, and
i-1
i
(iii) N = {e}.
n
A finite group G is solvable if and only if there exists a finite chain of subgroups G = N 
 0
N ... N such that
n
1
(i) N is a normal subgroup in N for i = 1, 2, . . ., n,
i
i-1
(ii) N / N is cyclic of prime order for i = 1, 2, . . ., n, and
i-1 i
(iii) N = {e}.
n
Let p be a prime number. Any finite p-group is solvable.

Let G be a group. An element g in G is called a commutator if

g = aba b
-1 -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’.
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
i-1
(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

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