8/3/2019 4(6)441test-02

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Math 4441/6441 (Spring 2008) Test #2 (04/08) Test Problems

Problem 1 (10 points). Let G be a group with |G| = 60 and let a G be a (fixed) elementsatisfying a42 = e, a16 = e and a15 = e. Determine ord(a). Show your work carefully.

Problem 2 (6+1+3 points). Let X = {1, 2, 3}. Consider S3 = {f1, f2, f3, f4, f5, f6} thatconsists of all bijective functions from X to X as listed below

f1 = 1X :1 1, 2 2, 3 3; f2 :1 1, 2 3, 3 2; f3 :1 2, 2 1, 3 3;

f4 :1 2, 2 3, 3 1; f5 :1 3, 2 1, 3 2; f6 :1 3, 2 2, 3 1.

It is known that S3 is a group under composition of functions.

(1) Let C(f3) = {g S3 | gf3 = f3g}. List all the distinct elements in C(f3) explicitly.(2) Let H = f6. List all the distinct elements in H explicitly and without repetition.(3) Determine the number of distinct left cosets of H = f6 in S3. For each left coset,

list all its elements explicitly in the format of f?H = {f?, . . . , f ?}.

Problem 3 (5+5 points). Let G be an abelian group and H be a (fixed) subgroup of G.

(1) Let F = {h2 | h H}. Show F is a subgroup of G.(2) Let T = {a G | ord(a) < }. Show T is a subgroup of G.

Problem 4 (4+4+2 points). Let G be a group. For any subsets H, K G, define HK ={hk | h H, k K}, which is clearly a subset of G.

(1) If (ab)2 = a2b2 for all a, b G, show G is abelian.(2) IfG is abelian, H G and K G (fixed subgroups), show HK G.(3) True or false: If H G and K G (but without the assumption that G is abelian),

then HK G. Give a proof or a counterexample. . . . . . . . . . . . . . True False

Problem 5 (10 points). True or false. Circle your choices and no justification is needed.Whenever G is mentioned, it is assumed that G is a group.

(1) Given a, b G, if ab = ba then ambn = bnam for all m, n Z. . . . . . . True False(2) Given a G, if a12 = e and a20 = e, then ord(a) = 4. . . . . . . . . . . . . . . True False

(3) Every group has at least two distinct subgroups. . . . . . . . . . . . . . . . . . . . True False(4) Given a G, if a175 = e, then ord(a) < . . . . . . . . . . . . . . . . . . . . . . . . True False(5) Given a, b G, if a = b, then a2 = b2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . True False(6) Given a, b G, if a = b, then a3 = b3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . True False(7) IfH1 G and H2 H1, then H2 G. . . . . . . . . . . . . . . . . . . . . . . . . . . . . True False(8) IfH1 G and H2 G, then H1 H2 G. . . . . . . . . . . . . . . . . . . . . . . . . True False(9) If |G| = 10, then there is no x G such that x6 = e. . . . . . . . . . . . . . . . True False

(10) Given a G, if a27 = e and a45 = e, then ord(a) = 27. . . . . . . . . . . . . . True False

For solutions, click here.

Math 4441 students need to do any four out of the five problems.Math 6441 students need to do all five problems.

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