Let . Check whether D is a UFD or not.
Observe that (D={f(x,y)+g(x,y)i \mid f,g\in \mathbb Z[x,y]}) can be written as (D=\mathbb Z[i][x,y]), since every element has coefficients in the Gaussian integers (\mathbb Z[i]), _____ ___ ________ _________ ________ _________ __________ _________ _____ ___ ______ ___.
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मान लीजिए कि G एक ऐसा समूह है कि चक्रीय है। सिद्ध कीजिए कि G आबेली होगा।
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.
.
(ग) मान लीजिए कि G कोटि वाला एक समूह है, जिसके केवल दो उपसमूह हैं -
और स्वयं G. G के लिए एक अल्पिष्ठ जनक समुच्चय ज्ञात कीजिए। साथ ही, यह भी ज्ञात कीजिए कि क्या n एक अभाज्य संख्या है, या भाज्य संख्या है, या दोनों में से कोई भी हो सकती है।
अतः, दर्शाइए कि यदि N, R की एक ऐसी गुणजावली है जिसका M एक उचित उपसमुच्चय है, तो होगा।
Give an example, with justification, of a commutative subgroup of a non-commutative group.
Find all the units of .
Check whether or not is a subgroup of
i) ,
ii)