# Exercise10 - b) Let C n = { e 2 πki n ∈ C | k ∈ Z }...

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Mathematics Department Mark Hamilton Room 7.548 V57 WT 2011/12 Advanced Higher Mathematics for INFOTECH Exercise sheet 10 1. a) Let G be a group such that a 2 = e for all elements a G . Prove that G is abelian. b) Let G be a group and H G a subgroup of index 2. Prove that H is normal. 2. a) Let G be a ﬁnite abelian group. If g G we write 2 g for the element g + g . Show that 2 X a G a = 0 . b) Find a ﬁnite abelian group G such that X a G a 6 = 0 . 3. Let U (1) = { e it C | t R } be the set of complex numbers of absolute value 1. a) Draw a picture of U (1) and prove that U (1) is with multiplication of complex numbers an abelian group.
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Unformatted text preview: b) Let C n = { e 2 πki n ∈ C | k ∈ Z } for n = 1 , 2 ,... . Draw a picture of C n for n = 1 ,..., 5 and prove that C n is a subgroup of U (1). c) Prove that C n is isomorphic to Z n = Z /n Z . 4. a) Determine the addition table (consisting of all sums of two elements) of the group Z 2 ⊕ Z 3 . b) Find the following elements in Z 17 = { , 1 ,..., 16 } : 39 ,-25 , 13 + 11 , 7-10 . c) Solve the following equations in Z 17 : 10-x = 4 , 7 + x = 2 ....
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## This note was uploaded on 01/18/2012 for the course INFORMATIK 2011 taught by Professor Phanthuongcang during the Winter '11 term at Cornell University (Engineering School).

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