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Unformatted text preview: Math 4107, exam 1
September 22, 2009
Each question is worth 20 points. 1. Deﬁne the following terms. a. Normal subgroup. b. Homomorphism. c. Innerautomorphism. 2. List all the elements of order 2 from the group S4 . 3. Find integers k and ℓ, satisfying 55k + 21ℓ = 1, |k | ≤ 20, |ℓ| ≤ 54. Don’t just write down the answer. Work it using the Euclidean algorithm (e.g. Knuth’s algorithm), so that I know you know some good mathematics. 4. Consider the set of all 2 × 2 matrices of the form 1a 01 . Prove that the set of all such matrices forms a group under matrix multiplication. 5. a. First, show that D4 has a normal subgroup of order 2 (in fact, it is a quite special type of normal subgroup...). Note that Dn for all n even has such a subgroup, and for n odd there is no such subgroup. b. Show that the group of automorphisms of the group D4 , denoted by Aut(D4 ), contains at least 4 elements. 1 ...
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