Math 5310, Exam 2, 23 September Name: 1. (8 points) Circle the correct answer. No justification needed for these. (a) Groups are cool true false (b) If g is an element of a group and gn = 1, then g...

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Math 5310, Exam 2, 23 September Name: 1. (8 points) Circle the correct answer. No justification needed for these. (a) Groups are cool true false (b) If g is an element of a group and gn = 1, then g has order n. true false (c) If every element g in a group G satisfies g4 = 1 then G is commutative. true false (d) Using cycle notation in S6, (5 2 4)(2 3 1 4)(6 5) = (5 6 2 3 1) true false 2. (4 points) Give an example a group homomorphism j ∶R+�→R× such that j(2) = 4. (Write down your ideas, I will give credit for partial answers and good ideas) 3. (8 points) Let G be a group with a trivial centre, i.e. Z(G) = {1}. Prove that if g ≠ h then cg ≠ ch, that is, prove that if g and h are different elements of G, then conjugation by g is not the same as conjugation by h. (Write down your ideas, I will give credit for partial answers and good ideas) 1 SOLUTIONS 0 Oo 0 Use 4K)=2× . This is a homomorphism since 2×-19 = 2×29 and 20=1 bgH : if Cg -- ch then gxg-t-hxh-ifhallx.ua rearrange : x g- ' = -g' hxh " so : Xj 'h= g-thx for all × So g- ' he ZCG ) -- fi } This means g- 'hi , and so g- h . It (extra space: clearly label anything here) 2