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intersection of normal subgroups is a normal subgroup

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If G is a finite group, define R = R(G) = INTERSECTION {K < G | G/K is solvable}.

a. Show that R is the smallest normal subgroup of G, such that G/R is solvable.
b. Show that G is solvable iff R = {1}.
c. If H <= G is a subgroup, show that R(H) <= H INTERSECTION R(G).

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Solution Preview

a. First of all, R as an intersection of normal subgroups is a normal subgroup. Let us show that G/R(G) is solvable. Let us denote the set of all normal subgroups of G such that G/K is ...

Solution Summary

Intersections of normal subgroups are exemplified.