Final Exam Problems for Math 4441/6441 | Exams Algebra

Math 4441/6441 (Spring 2008) Final Exam (04/29) Test Problems

Problem 1 (10 points).Let Gbe a group with |G|= 90 and let a∈Gbe a (fixed) element.

(1) If a72 =e, determine all the possible value(s) of ord(a).

(2) If a72 =e,a60 =e,a21 6=eand a20 6=e, determine ord(a). Show your reasoning.

Problem 2 (10 points).Consider the group Z9(under addition) and also consider the group

U(Z9) (under multiplication).

(1) In Z9, determine ord([6]) and h[6]i.

(2) In U(Z9), determine ord([4]) and h[4]i.

(3) In U(Z9), determine ord([8]) and h[8]i.

(4) Determine whether U(Z9) is a cyclic group. If so, find a generator of U(Z9).

Problem 3 (10 points).Determine whether each of the following is a group homomorphism

and, if so, determine whether it is an isomorphism.

(1) Let Gbe a group and a∈Gbe a (fixed) element. Define f:G→Gby f(x) = axa−1

for all x∈G.

(2) Consider groups Z2(under addition) and U2={1,−1}(under multiplication). Define

g:Z2→U2by g([0]) = 1 and g([1]) = −1.

(3) Consider groups Z2(under addition) and U4={1, i, −1,−i}(under multiplication).

Define h:Z2→U4by h([0]) = 1 and h([1]) = i.

Problem 4 (10 points).Let Gand Hbe groups, ϕ:G→Hbe a group homomorphism,

a∈Gand ϕ(a) = b. Denote by eGand eHthe identity elements of Gand Hrespectively.

(1) If an=eGfor some n∈Z, show bn=eH(for the same n).

(2) If ord(a) = k < ∞, show ord(b)<∞.

(3) More explicitly, if ord(a) = 22, then what are all the possible value(s) of ord(b)?

Problem 5 (10 points).Let Gand Hbe groups, f:G→Hbe a group homomorphism.

Also let aand bbe (given) elements in Gand denote f(a) = xand f(b) = y.

(1) Show that xy =yx if and only if a−1b−1ab ∈Ker(f).

(2) Show that f(a) = f(b)if and only if a=bk for some k∈Ker(f).

Problem 6 (10 points).Let Gand Hbe groups, f:G→Hbe a group homomorphism

and K≤Hbe a (fixed) subgroup of H. Let f−1(K) = {a∈G|f(a)∈K}.

(1) Show f−1(K)≤G, i.e., show f−1(K) is a subgroup of G.

(2) Further assume KH. Show f−1(K)G.

Problem 7 (10 points).True or false. Circle your choices and no justification is needed.

Whenever Gis mentioned, it is assumed that Gis a group.

(1) There is only one homomorphism from S3to (Z,+). ...............True False

(2) There is only one homomorphism from (Z,+) to S3. ...............True False

(3) Every group has at least one normal subgroup. . . . . . . . . . . . . . . . . . . . . True False

(4) There are at least two distinct homomorphisms from Gto G. ......True False

(5) Given a∈G, if am=e=anwith m, n ∈Z, then agcd(m,n)=e. . . . . . True False

(6) Given a, b ∈Gwith |G|= 21, if a6=b, then a26=b2. ...............True False

(7) Given a, b ∈Gwith |G|= 21, if a3=b3, then a=b. ...............True False

(8) If H1, H2, K ≤Gand H16=H2, then H1K6=H2K. ................True False

(9) If HGand KG, then HK G. ..............................True False

(10) Given a∈G, if a54 =eand a45 =e, then ord(a)=9. ..............True False

For solutions, click here.

Math 4441 students need to do any five out of the seven problems.

Math 6441 students need to do any six out of the seven problems.

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Download Final Exam Problems for Math 4441/6441 and more Exams Algebra in PDF only on Docsity!

Math 4441/6441 (Spring 2008) Final Exam (04/29) Test Problems

Problem 1 (10 points). Let G be a group with |G| = 90 and let a ∈ G be a (fixed) element.

(1) If a^72 = e, determine all the possible value(s) of ord(a). (2) If a^72 = e, a^60 = e, a^21 6 = e and a^20 6 = e, determine ord(a). Show your reasoning.

Problem 2 (10 points). Consider the group Z 9 (under addition) and also consider the group U (Z 9 ) (under multiplication).

(1) In Z 9 , determine ord([6]) and 〈[6]〉. (2) In U (Z 9 ), determine ord([4]) and 〈[4]〉. (3) In U (Z 9 ), determine ord([8]) and 〈[8]〉. (4) Determine whether U (Z 9 ) is a cyclic group. If so, find a generator of U (Z 9 ).

Problem 3 (10 points). Determine whether each of the following is a group homomorphism and, if so, determine whether it is an isomorphism.

(1) Let G be a group and a ∈ G be a (fixed) element. Define f : G → G by f (x) = axa−^1 for all x ∈ G. (2) Consider groups Z 2 (under addition) and U 2 = { 1 , − 1 } (under multiplication). Define g : Z 2 → U 2 by g([0]) = 1 and g([1]) = −1. (3) Consider groups Z 2 (under addition) and U 4 = { 1 , i, − 1 , −i} (under multiplication). Define h : Z 2 → U 4 by h([0]) = 1 and h([1]) = i.

Problem 4 (10 points). Let G and H be groups, ϕ : G → H be a group homomorphism, a ∈ G and ϕ(a) = b. Denote by eG and eH the identity elements of G and H respectively.

(1) If an^ = eG for some n ∈ Z, show bn^ = eH (for the same n). (2) If ord(a) = k < ∞, show ord(b) < ∞. (3) More explicitly, if ord(a) = 22, then what are all the possible value(s) of ord(b)?

Problem 5 (10 points). Let G and H be groups, f : G → H be a group homomorphism. Also let a and b be (given) elements in G and denote f (a) = x and f (b) = y.

(1) Show that xy = yx if and only if a−^1 b−^1 ab ∈ Ker(f ). (2) Show that f (a) = f (b) if and only if a = bk for some k ∈ Ker(f ).

Problem 6 (10 points). Let G and H be groups, f : G → H be a group homomorphism and K ≤ H be a (fixed) subgroup of H. Let f −^1 (K) = {a ∈ G | f (a) ∈ K}.

(1) Show f −^1 (K) ≤ G, i.e., show f −^1 (K) is a subgroup of G. (2) Further assume K H. Show f −^1 (K) G.

Problem 7 (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) There is only one homomorphism from S 3 to (Z, +)................ True False (2) There is only one homomorphism from (Z, +) to S 3................ True False (3) Every group has at least one normal subgroup..................... True False (4) There are at least two distinct homomorphisms from G to G....... True False (5) Given a ∈ G, if am^ = e = an^ with m, n ∈ Z, then agcd(m,n)^ = e...... True False (6) Given a, b ∈ G with |G| = 21, if a 6 = b, then a^2 6 = b^2................ True False (7) Given a, b ∈ G with |G| = 21, if a^3 = b^3 , then a = b................ True False (8) If H 1 , H 2 , K ≤ G and H 1 6 = H 2 , then H 1 K 6 = H 2 K................. True False (9) If H G and K G, then HK G............................... True False (10) Given a ∈ G, if a^54 = e and a^45 = e, then ord(a) = 9............... True False

For solutions, click here. Math 4441 students need to do any five out of the seven problems. Math 6441 students need to do any six out of the seven problems. 1

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