जाँच कीजिए कि निम्नलिखित का एक उपसमूह है या नहीं :
A = { z ∈ ℂ ∣ |z| ∈ ℚ } को जाँचना है कि यह किसी दिए गए समुच्चय का उपसमुच्चय है या नहीं। मान लेते हैं z = a + ib जहाँ a, b ________ _____ __________ _____ _________ ______ _____ ______ _________ _________ _________.
_____ _______ ________ ________ _____ _________.
______ ________ ______ _______ __________ __________ _____ _________ ________ ______ _____.
_______ _________ _________ _________ _________ ___.
___ ______ ________ ______ __________ __________ _________ ___ ______ __________.
____ ________ ____ ______ ______ ______ _________ ___ __________ _________.
___ _________ _______ ___ ___ _______ ___ ______ ________ ____.
____ ___ _______ _____ ____ ________ ____ ________ _______ _______ ______ __________.
_____ ________ ___ ____ __________ _____ ______ ____ ________ __________.
________ _______ _________ _______ ________ ________ ________ ___ ________ _____ _____ ____.
_______ _______ _____ _______ ____.
__________ ________ ____ ___ ________ __________ ____ _________.
____ ________ ___ ________ ______ _____ __________ _______ _________ __________.
_____ __________ __________ _______ ______.
________ _______ _________ _______ __________ _____ _____ __________ ________ ___ _____ _________.
__________ _________ _________ _____ ____ _________ ______ _____ ________.
Get Full Answer on WhatsApp
Consider the map . Let
. Then B is a group with respect to the composition of functions. Check whether or not
is a normal subgroup of B.
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 , with only two subgroups —
and itself. Find a minimal generating set for G. Also, find out whether n is a prime or a composite number, or can be either.
Let be a finite abelian group and
. Prove that
.
मान लीजिए कि . जाँच कीजिए कि D एक UFD है या नहीं।
(क) फलन पर विचार कीजिए। मान लीजिए कि
. तब, फलनों के संयोजन के सापेक्ष B एक समूह है। जाँच कीजिए कि
, B का एक प्रसामान्य उपसमूह है या नहीं। (4)
र्शाइए कि यदि या
, तो
, जहां
.