By Mark Steinberger

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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. (a) Let H ⊂ G be a subgroup. Deﬁne f −1 (H ) ⊂ G by f −1 (H ) = {x ∈ G | f (x) ∈ H }. Show that f −1 (H ) is a subgroup of G which contains ker f . (b) Let H ⊂ G be a subgroup. Deﬁne f (H) ⊂ G by f (H) = {f (x) | x ∈ H}. Show that f (H) is a subgroup of G . If ker f ⊂ H, show that f −1 (f (H)) = H. (c) Suppose that H ⊂ im f . Show that f (f −1 (H )) = H . Deduce that there is a one-to-one correspondence between the subgroups of im f and those subgroups of G that contain ker f .

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 ). We denote this group by G × H. 2. The direct product G × H is a group.

For m, m ∈ M , f (mm ) = f (m)f (m ). An isomorphism of monoids is a homomorphism f : M → M which is bijective. 21. 1. Show that any group with one element is isomorphic to the trivial group. 2. Show that any group with two elements is isomorphic to Z2 . 3. 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.