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Unformatted text preview: the dodecahedron to the icosahedron and the the tetrahedron to itself.) 4. Let G = g with | G | = n and G = g with | G | = m . Show that G G is cyclic if and only if ( m,n ) = 1 . 5. Let G be a nite 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 A B . Give an example where A B with A = B but A B . 1 8. Let G be a group, x G and y C G ( x ) . Show that x,y 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 n-1 1 (mod p n ) but (1 + p ) p n-2 is not equivalent to 1 (mod p n ). Dedude that 1 + p is an element of order p n-1 in the multiplicative group ( Z /p Z ) . 2...
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