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# hw19 - Exercise 11 Let G be a ﬁnite group let H ≤ G and...

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Modern Algebra 1 Homework 8.3 Spring 2010 Due March 24 Exercise 10. Let f : R × R R be given by f ( x, y ) = 2 x - 3 y . a. Prove that f is a surjective homomorphism. b. Find ker f and describe it and its cosets geometrically. c. The First Isomorphism Theorem implies that f induces an isomorphism f : ( R × R ) / ker f R . Describe this isomorphism geometrically.
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Unformatted text preview: Exercise 11. Let G be a ﬁnite group, let H ≤ G and let K C G . Prove that if | H | is relatively prime to [ G : K ] then H ≤ K . [ Hint: Given a ∈ H , by considering the orders of a in H and aK in G/K , show that h a i ≤ K .] Exercise 12. Lang, II.4.30....
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