19. Prove that the group (Z/11Z)* is cyclic, and find a generator.
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- Let H1={ [ 0 ],[ 6 ] } and H2={ [ 0 ],[ 3 ],[ 6 ],[ 9 ] } be subgroups of the abelian group 12 under addition. Find H1+H2 and determine if the sum is direct.If G is a cyclic group, prove that the equation x2=e has at most two distinct solutions in G.9. Find all homomorphic images of the octic group.
- Let A={ a,b,c }. Prove or disprove that P(A) is a group with respect to the operation of union. (Sec. 1.1,7c)27. a. Show that a cyclic group of order has a cyclic group of order as a homomorphic image. b. Show that a cyclic group of order has a cyclic group of order as a homomorphic image.9. Suppose that and are subgroups of the abelian group such that . Prove that .
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