Algebra I: Section 6. The structure of groups. 6.1 Direct products of ...

and the middle group is solvable if the end groups are (Lemma 6.4.9). Thus a group G always
contains a unique largest solvable normal subgroup R, which is sometimes referred to as the
solvable radical of G. The quotient G/R contains no normal solvable subgroups at all,
although there will certainly be non-normal abelian and solvable subgroups in it, for instance
cyclic subgroups.
6.4.16 Exercise. For any n ≥ 3 determine the center Z(G) of the dihedral group G = Dn.
Hint: The answer will depend on whether n is even or odd. �
6.4.17 Exercise. Compute the commutator subgroup [G, G] for the dihedral group G =
Dn, n ≥ 3. Is Dn nilpotent for any n? Solvable? �
6.4.18 Exercise. In the cases where the center of Dn is nontirival, does the extension
e → Z(Dn) → Dn → Dn/Z(Dn) → e
split – i.e. is Dn a semidirect product with its center as the normal subgroup? �
6.4.19 Exercise. Compute the commutator subgroup [G, G] of the quaternion group G = Q8
of Example 6.3.10. Is Q8 nilpotent? Solvable? �
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