By Mark Steinberger

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**Sample text**

Now let us generalize the material in this section to the study of products of more than two groups. 9. Suppose given groups G1 , . . , Gk , for k ≥ 3. The product (or direct product), G1 × · · · × Gk , of the groups G1 , . . , Gk is the set of all k-tuples (g1 , . . , gk ) with gi ∈ Gi for 1 ≤ i ≤ k, with the following multiplication: (g1 , . . , gk ) · (g1 , . . , gk ) = (g1 g1 , . . , gk gk ). The reader should be able to supply the proof of the following proposition. 7 Of course, neither H nor K need be abelian.

List the homomorphisms from Z9 to Z6 . 4. List the homomorphisms from Z5 to Z6 . 5. Show that the order of m in Zn is n/(m, n). Deduce that the order of each element divides the order of Zn . Deduce that every non-identity element of Zp has order p, for any prime p. † 6. Let G be a group and let x ∈ G. Show that o(x−1 ) = o(x). ) CHAPTER 2. GROUPS: BASIC DEFINITIONS AND EXAMPLES 34 7. Show that a group G is cyclic if and only if there is a surjective homomorphism f : Z → G. † 8. Let f : G → G be a homomorphism.

Show that every subgroup of Q8 other than e contains a2 . ) 9. Let n be any power of 2. Show that every subgroup of Q4n other than e contains a2 . 10 Direct Products We give a way of constructing new groups from old. It satisﬁes an important universal property with respect to homomorphisms. 1. Let G and H be groups. Then the product (or direct product) of G and H is the group structure on the cartesian product G × H which is given by the binary operation (g1 , h1 ) · (g2 , h2 ) = (g1 g2 , h1 h2 ).