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

-
: a , b R, a two b2 0
b a
a b be a group under
multiplication and C* be a group of non zero complex numbers under
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multiplication. Show that f : C* ? G described by f(a + ib) = ??
?
??
?
- b a
a b , for
all a + ib ? C*, is an isomorphism.
6) describe a group homomorphism. Prove that homomorphic image of a
cyclic group is cyclic.
7) Let f : G ? G' be a group homomorphism. Prove that
i) f(e) is the identity element of G', where e is the identity element
of G
ii) f(a-1) = (f(a))-1, for all a ? G
iii) f(am) = (f(a))m, for all a ? G, m ? Z.
8) Let (C* , ??) .(R* , ??) be groups of non zero complex numbers, non zero real
numbers respectively under multiplication. Show that f : C* ? R* described
by f(z) = | z |, for all z ? C*, is a group homomorphism. obtain Ker(f). Is f
onto? Why?
9) Let (C* , ??) , (R* , ??) be groups of non zero complex numbers, non zero
real numbers respectively under multiplication. Show that f : C* ? R*
described by f(z) = | z |, for all z ? C*, is a group homomorphism. obtain
Ker(f). Is f onto? Why?
10) Let G = {1 , -1} be a group under multiplication. Show that f : (Z , +) ?
G described by f(n) =
? ? ? -
1
,
if n
is odd
1 , if n iseven
is onto group homomorphism. obtain Ker(f).
11) Let (R+ , ??) be a group of positive reals under multiplication. Show that f :
(R , +) ? R+ described by f(x) = 2x, for all x ? R, is an isomorphism.
12) Let (R+ , ??) be a group of positive reals under multiplication. Show that f :
(R , +) ? R+ described by f(x) = ex, for all x ? R, is an isomorphism.
13) If f : G ? G' is an isomorphism and a ? G then show that o(a) = o(f(a)).
14) Prove that every finite cyclic group of order n is isomorphic to (Zn , +n).
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15) Prove that every infinite cyclic group is isomorphic to (Z , +).
16) Let G be a group of all non singular matrices of order two over the set of
reals and R* be a group of all nonzero reals under multiplication. Show
that f : G ? R* , described by f(A) = | A |, for all A ? G, is onto group
homomorphism. Is f 1 one? Why?
17) Let G be a group of all non singular matrices of order n over the set of
reals and R* be a group of all nonzero reals under multiplication. Show
that f : G ? R* , described by f(A) = | A |, for all A ? G, is onto group
homomorphism.
18) Let R* be a group of all nonzero reals under multiplication. Show that f :
R* ? R* , described by f(x) = | x |, for all x ? R*, is a group
homomorphism. Is f onto? Justify.
19) Prove that every group is isomorphic to it self. If G1 , G2 are groups such
that G1 ? G2 then prove that G2 ? G1.
20) Let G1 , G2 , G3 be groups such that G1 ? G2 and G2 ? G3. Prove that
G1 ? G3.
21) Show that f : (C , +) ? (C , +)defined by f(a + ib) = –a + ib, for all a + ib
? C, is an automorphism.
22) Show that f : (C , +) ? (C , +) described by f(a + ib) = a – ib, for all a + ib
? C, is an automorphism.
23) Show that f : (Z , +) ? (Z , +) described by f(x) = – x, for all x ? Z, is an
automorphism.
24) Let G be an abelian group. Show that f : G ? G described by f(x) = x-1, for
all x ? G, is an automorphism.
25) Let G be a group and a ? G. Show that fa : G ? G described by fa(x) =
axa-1, for all x ? G, is an automorphism.
26) Let G be a group and a ? G. Show that fa : G ? G described by fa(x) =
a-1xa, for all x ? G, is an automorphism.
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27) Let G = {a , a2 , a3 , - - - , a12 (= e)}be a cyclic group generated by a.
Show that f : G ? G described by f(x) = x4, for all x ? G, is a group
homomorphism. obtain Ker(f).
28) Let G = {a , a2 , a3 , - - - , a12 (= e)}be a cyclic group generated by a.
Show that f : G ? G described by f(x) = x3, for all x ? G, is a group
homomorphism. obtain Ker(f).
29) Show that f : (C , +) ? (R , +) described by f(a + ib) = a, for all a + ib ?
C, is onto homomorphism. obtain Ker(f).
30) Show that homomorphic image of a finite group is finite. Is the converse
true? Justify.

Earning:   Approval pending.