MAT-3110 Final: MATH 3110 App State Fall2009 Final Exam answer key

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turquoisegnu128

15 Feb 2019

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December 9th, 2009: (12 points) for each of the following pairs of groups, if the groups are isomorphic, circle g1 = g2 and explain why they are isomorphic. If the groups aren"t isomorphic, circle g1 6 = g2 and explain why not. (a) q 6 = q [where q = rationals, q = quaternions] There are pleanty of reasons why the rationals and quaternions cannot be isomorphic. The easiest reasons are (1) the rationals are an in nite group while the quarternions are a group of order 8 and (2) the rationals are an abelian group while the quaternions are not. But there are other (harder & sillier) ways to see that these groups aren"t isomorphic. For example, all of the non-identity elements in q have in nite order (0 6= r q then r + r + + r = nr 6= 0 unless n = 0).

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These questions are asking about subrgoups, groups, isormorphic.

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(below question)

SAMPLE MIDTERM 1, MATH 120A

(1) 1. Write the denition of group.

2. Give an example of a group of order 2011.

3. Give an example of a non-abelian group with 12 elements.

(2) Let (G,*) and (H . ) be groups.

1. Dene what it means for G and H to be isomorphic.

2. Assume that G has exactly two solutions to the equation x^2 =e.

Prove, directly from the denition in (1), that if G and H areiso-

morphic, then H has exactly two solutions to the equation x^2 =e.

3. Prove or disprove: (R, +) is isomorphic (R+, . ). Here R+ ={x is subset of R : x>0}.

4. Prove or disprove: (R, +) is isomorphic (R*, . ). Here R* ={x is subset of R : x is not 0}

(3) Let G be the group Z_20, with addition modulo 20.

1. Draw the subgroup diagram of G, giving a generator for eachsubgroup

2. Find the subgroup H =< 16 > of G.

3. Find a subgroup K of U_20 isomorphic to H.

(4) Let (G, . ) be a group and let H be a subgroup of H. Fix anelement a of

G and let K = { aha^-1 : h is subset of H}

1. Prove that K is a subgroup of G.

2. Prove that the map psi : H -> K, h -> aha^-1 isan isomorphism of groups.

(5) Let (G, . ) be a group and let a be an element of G.

1. Dene the order of a.

2. Give an example of an element of order 4 in GL(2,R).

3. Find the order of 16 in Z_20.

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MAT-3110 Final: MATH 3110 App State Fall2009 Final Exam answer key

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