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

15) Let f : G ? G' be a group homomorphism and o(a) is finite, for all a ? G. If f
is 1 one then show that o(f(a)) = o(a).
16) Let f : G ? G' be a group homomorphism and o(f(a)) = o(a), for all a ? G.
Show that f is 1 one.
2 : Multiple option ques. of one marks
select the accurate choice from the provided choices.
1) Every finite cyclic group of order n is isomorphic to - - -
a) (Z , +) b) (Zn , +n) c) (Zn , × n) d) (Z'
n , × n)
2) Every infinite cyclic group is isomorphic to - - -
a) (Z , +) b) (Zn , +n) c) (Zn , × n) d) (Z'
n , × n)
3) Let f : G ? G' be a group homomorphism and a ? G. If o(a) is finite
then - - -
a) o(f(a)) = eight b) o(f(a))?o(a).
c) o(a)?o(f(a)) d) o(f(a)) = 0.
4) A group G = {1 , -1 , i , -i} under multiplication is not isomorphic to -
- -
a) (Z4 , +4) b) G
c) (Z'
8 , × 8) d) none of these.
5) Let f : G ? G' be a group homomorphism. If G is abelian then f(G) is
- - -
a) non abelian b) abelian
c) cyclic d) empty set
19
6) Let f : G ? G' be a group homomorphism. If G is cyclic then f(G) is -
- -
a) non abelian b) non cyclic
c) cyclic d) finite set
7) A onto group homomorphism f : G ? G' is an isomorphism if Ker(f) =
- - -
a) f b) {e) c) {e'} d) none of these
8) A function f : G ? G , (G is a group) , described by f(x) = x-1, for all x
? G, is an automorphism if and only if G is - - -
a) abelian b) cyclic c) non abelian d) G = f.
3 : ques. of four marks
1) Let f : G ? G' be a group homomorphism . prove that f(G) is a subgroup
of G'. Also prove that if G is abelian then f(G) is abelian.
2) Let f : G ? G' be a group homomorphism. Show that f is 1 one if and
only if Ker(f) = {e}.
3) Let G = {1 , -1 , i , -i} be a group under multiplication. Show that f : (Z ,
+) ? G, described by f(n) = in , for all n ? Z, is onto group homomorphism.
obtain Ker(f).
4) Let G = {1 , -1 , i , -i} be a group under multiplication. Show that f : (Z ,
+) ? G, described by f(n) =(–i)n , for all n ? Z, is onto group
homomorphism. obtain Ker(f).
5) Let G =
? ? ?
? ? ?
? + ? ??
?
??
?

Earning:   Approval pending.