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Unformatted text preview: exercise 12 that R / Z is isomorphic to the circle group S 1 , i.e. the subgroup of C * consisting of elements of norm 1, then look at the torsion subgroups of both sides), 15, 36*, 42 Section 3.2 #9*, 20*, 21 Section 3.3 #3, 7* (suggestion: use the ﬁrst isomorphism theorem). Exercises not from the text: (all to be handed in): 1*. Suppose that G = H ∪ K , where H and K are subgroups of G . Show that either G = H or G = K (i.e., no group is the union of two proper subgroups.) 1 2*. Suppose that G is a ﬁnite group and that G = H 1 ∪ H 2 ∪ H 3 , where each H i is a proper subgroup of G . Show that  G : H i  = 2 for all i . Also, ﬁnd an example where this actually happens. (Hint: ﬁrst show by counting that at least one of the subgroups, say H 1 , has index 2. Then prove that this forces H 1 H i = G and  H i : H 1 ∩ H i ] = 2, for i = 2 , 3, and do some more counting.) 2...
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 Fall '08
 Bunch,J
 Math, Group Theory, Cyclic group, Group isomorphism, Dummit

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