Question

31. Show that an intersection of normal subgroups of a group G is again a normal subgroup of G. 32. Given any subset S of a group G, show that it makes sense to speak of the smallest normal subgroup that contains S. [Hint: Use Exercise 31.] 33. Let G be a group. An element of G that can be expressed in the form aba-1b-1 for some a, b ? G is a commutator in G. The preceding exercise shows that there is a smallest normal subgroup C of a group G containing all commutators in G; the subgroup C is the commutator subgroup of G. Show that G/C is an abelian group.

          31. Show that an intersection of normal subgroups of a group G is again a normal subgroup of G.
32. Given any subset S of a group G, show that it makes sense to speak of the smallest normal subgroup that
contains S. [Hint: Use Exercise 31.]
33. Let G be a group. An element of G that can be expressed in the form aba-1b-1 for some a, b ? G is a
commutator in G. The preceding exercise shows that there is a smallest normal subgroup C of a group G
containing all commutators in G; the subgroup C is the commutator subgroup of G. Show that G/C is an
abelian group.

31. Show that an intersection of normal subgroups of a group G is again a normal subgroup of G.
32. Given any subset S of a group G, show that it makes sense to speak of the smallest normal subgroup that
contains S. [Hint: Use Exercise 31.]
33. Let G be a group. An element of G that can be expressed in the form aba-1b-1 for some a, b ? G is a
commutator in G. The preceding exercise shows that there is a smallest normal subgroup C of a group G
containing all commutators in G; the subgroup C is the commutator subgroup of G. Show that G/C is an
abelian group.

Added by Juan Francisco J.

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