HOMEWORK 2, DUE WED. SEP 17.
Sec. 2.5 Ex. 2.47, 2.57, 2.59, 2.60, 2.61
Sec. 2.6 Ex. 2.65, 2.66, 2.71, 2.76
Additional Problems:
1. Let Gbe the dihedral group D2ngiven by G={xiyj:x2=e, yn=
e, xy =y−1x}.
(a) Show that the subgroup N={e, y, . . . , yn−1}is normal in G.
(b) Show that G/N ∼
=Z/2Z.
2. Let G=Runder addition and N=Z. Show that G/N ' {z∈C×,|z|=
1}, the unit circle in the complex plane.
3. Classify all groups of order 8.
4. In an abelian group all subgroups are normal. Is the converse true ? i.e.
if Gis a group in which all subgroups are normal, does Ghave to be abelian
? If yes, prove this; if not, give a counterexample.
5. Give an example of a group G, subgroup Hand an element a∈Gsuch
that a−1Ha ⊂H, but a−1H a 6=H.
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