के सभी मात्रकों को ज्ञात कीजिए।
(\mathbb{Z}_{12}={0,1,2,3,4,5,6,7,8,9,10,11}) में वे तत्त्व मात्रक होते हैं जिनका (\gcd(a,12)=1) हो। (\gcd(0,12)=12) अतः 0 मात्रक नहीं है। (\gcd(1,12)=1) अतः 1 मात्रक है। _______ _____ _____ _____ _______ ___.
__________ ______ _____ ________ _______ _______.
____ ________ _______ ______ _______ ________.
____ _____ __________ ____ _____ ________ _______ ____ _______.
________ _______ ____ ____ ___ ___ ____ ____.
______ ______ ________ _______ ___ _______ ________ _____ ___ ___ __________ ___.
______ _______ _________ ________ ________ _________ _________ ______ ______ _____ _________.
______ ____ ________ ________ _______ _______.
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Let and
सिद्ध कीजिए कि (n बार)
और
.
Which of the following statements are true? Justify your answers. (This means that if you think a statement is false, give a short proof or an example that shows it is false. If it is true, give a short proof for saying so.) (20)
i) , where
is the Euler-phi function.
ii) If G1 and G2 are groups, and is a group homomorphism, then
.
iii) If G is an abelian group, then G is cyclic.
iv) If G is a group and , then
.
v) Every element of Sn has order at most n.
vi) If R is a ring and I is an ideal of R, then and
.
vii) If is a product of an even number of disjoint cycles, then
.
viii) If a ring has a unit, then it has only one unit.
ix) The characteristic of a finite field is zero.
x) The set of discontinuous functions from [0, 1] to form a ring with respect to point-wise addition and multiplication.
Let G be a group of order 56. What are all its Sylow p-subgroups? Show that G is not simple, i.e., G must have a proper normal non-trivial subgroup.
Check whether or not is a subgroup of
i) ,
ii)
Prove that and
.
सिद्ध कीजिए कि वलयों के रूप में, .