North Maharashtra University 2008 B.Sc Mathematics S.YBSc MTH – 212 (B) (Computational Algebra) - exam paper

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NORTH MAHARASHTRA UNIVERSITY, JALGAON

(New Syllabus w.e.f. June 2008)
Class : S.Y. B. Sc. Subject : Mathematics
Paper : MTH – 212 (B) (Computational Algebra)
(Calculus of Several Variables)

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1 : ques. of two marks
1) Let (R , +) be a group of real numbers under addition. Show that f : R ? R,
described by f(x) = 3x , for all x ? R, is a group homomorphism. obtain Ker(f).
2) Let (R , +) be a group of real numbers under addition. Show that f : R ? R,
described by f(x) = 2x , for all x ? R, is a group homomorphism. obtain Ker(f).
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3) If (R , +) is a group of real numbers under addition and (R+ , ??) is a group of
positive real numbers under multiplication. Show that f : R ? R+, described by
f(x) = ex , for all x ? R, is a group homomorphism. obtain Ker(f).
4) Let (R* , ??) be a group of non zero real numbers under multiplication. Show
that f : R* ? R*, described by f(x) = x3 , for all x ? R*, is a group
homomorphism. obtain Ker(f).
5) Let (C* , ??) be a group of non zero complex numbers under multiplication.
Show that f : C* ? C*, described by f(z) = z4 , for all z ? C*, is a group
homomorphism. obtain Ker(f).
6) Let (Z , +) be a group of integers under addition and G = {5n : n ? Z} a group
under multiplication. Show that f : Z ? G, described by f(n) = 5n , for all n ? Z,
is onto group homomorphism.
7) Let (Z , +) and (E , +) be the groups of integers and even integers respectively
under addition. Show that f : Z ? E, described by f(n) = 2n , for all n ? Z, is an
isomorphism.
8) describe a group homomorphism. Let (G , *) , (G' , *') be groups with identity
elements e , e' respectively. Show that f : G ? G', described by f(x) = e' , for all
x ? G, is a group homomorphism.
9) Let G = {a , a2 , a3 , a4 , a5 = e} be the cyclic group generated by a. Show that
f : (Z5 , +5) ? G, described by f( n ) = an , for all n ? Z5, is a group
homomorphism. obtain Ker(f).
10) Let f : (R , +) ? (R , +) be described by f(x) = x + one , for all x ? R. Is f a group
homomorphism? Why?
11) Let G = {1 , -1 , i , -i} be a group under multiplication and Z'
8 = {1 , three , five ,
7 } a group under multiplication modulo 8. Show that G and Z'
8 are not
isomorphic.
12) Show that the group (Z4 , +4) is isomorphic to the group (Z'
5 , × 5).
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13) Let f : G ? G' be a group homomorphism. If a ? G and o(a) is finite then
show that o(f(a))?o(a).
14) Let f : G ? G' be a group homomorphism If H' is a subgroup of G' then
show that Ker(f) ? f -1(H').

Earning:   Approval pending.