Q₁ Let H be a subgroup of G of orders (s is a fixed number). Show that the intersection of all Subgroups of G of orders is a normal subgroup of G. Q2: Let C be the Commutator subgroup of G. (1) Show that every element of C is a finite Product of Commutators (2) Use part(1) to show that C 4G. : Let C be the Commutator subgroup of G. Prove that if NAG then G/N is abelian C≤N. Q4: Let Ay be the alternating subgroup of the group of permutations Sy. Show that A4 Contains no subgroup of order 6. Dago. A group G is called a simple group if the only normal subgroups of G are {ez and G. . Def. Let MAG such that M+G. Then M is called a maximal normal subgroup of G, if for any normal subgroup. IN of G with MCN, then N=M or N = G. Q5: Irove that M is a maximal normal subgroup of GE G/M is simple.