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Unformatted text preview: the dodecahedron to the icosahedron and the the tetrahedron to itself.) 4. Let G = é g ê with  G  = n and G Í = é g Í ê with  G Í  = m . Show that G × G Í is cyclic if and only if ( m,n ) = 1 . 5. Let G be a ﬁnite cyclic group with  G  = p where p is prime. Show that G has no proper subgroups. 6. Show that Q × Q is not cyclic. 7. Section 2.4, Exercise 2: Prove that if A is a subset of B then é A ê ≤ é B ê . Give an example where A ⊂ B with A Ó = B but é A ê ≤ é B ê . 1 8. Let G be a group, x ∈ G and y ∈ C G ( x ) . Show that é x,y ê is abelian. 9. Section 2.3, Exercise 21: Let p be an odd prime and n be a positive integer. Use the Binomial Theorem to show that (1 + p ) p n1 ≡ 1 (mod p n ) but (1 + p ) p n2 is not equivalent to 1 (mod p n ). Dedude that 1 + p is an element of order p n1 in the multiplicative group ( Z /p Z ) × . 2...
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This note was uploaded on 01/21/2012 for the course MATH 113 taught by Professor Ogus during the Fall '08 term at Berkeley.
 Fall '08
 OGUS
 Math, Algebra

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