# Exam1 - G . Prove that HN = { hn : h H,n N } is a subgroup...

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MATH 373: Exam 1 Monday, February 15, 2010 Answer all questions in the answer booklet. Please do these in order. You are expected to justify your answers in a manner that an average MATH 373 student should be able to follow. This includes sentences for clariﬁcation. Point values are on the right hand side. They (should) sum up to 100. 1. Determine if the following sets G with the operation given form a group. Justify your answers. That is, if G is a group, show that it satisﬁes all the group axioms. If G is not a group, show that it fails to satisfy at least one axiom. [21 pts] (a) G is the set of all rational numbers, and for x,y G , x * y = xy . (b) G is the set of all rational numbers x 6 = 0, and for x,y G , x * y = x/y . (c) G is the set of all real numbers x 6 = - 1 and for x,y G , x * y = x + y + xy . 2. If f : X Y is a function and B Y is a subset then the inverse image of B is deﬁned as f - 1 ( B ) = { x X : f ( x ) B } . If f : G H is a homomorphism and B H show that f - 1 ( B ) G . [10 pts] 3. Let G be a group, let H be a subgroup of G , and let N be a normal subgroup of
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Unformatted text preview: G . Prove that HN = { hn : h H,n N } is a subgroup of G . [10 pts] 4. (a) Let n 5. Give an Abelian subgroup of S n of order 5. [5 pts] (b) Write (3541)(6312)-1 (52) in both disjoint cycle notation and as a product of 2-cycles. [5 pts] (c) Find an element in S 9 of order 12 which does not x any number. [5 pts] (d) Give 2 non-disjoint cycles in S 5 whose product is not a cycle. [5 pts] 5. Let G = ( Z 15 , +) and H G such that H = h 3 i . List the distinct left cosets of H in G . [7 pts] 6. Prove that ( R \ { } , ) is not cyclic (do not use | R | as your argument). [12 pts] 7. Show that the order of an element is isomorphism invariant. (i.e. If : G = H then o ( g ) = o ( ( g ))) [10 pts] 8. Show that if H G with the property that gH = Hg for all g G then H must be normal in G . (note: Hg = { hg : h H } ) [10 pts]...
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