समूह , के अवयवों और इसकी संरचना को स्पष्ट रूप में दीजिए।
समूह (S_n / A_n), जहाँ (n \geq 5), का विश्लेषण करने के लिए पहले हमें (S_n) और (A_n) की परिभाषा याद करनी होगी। (S_n) सभी (n) तत्वों के क्रमचयन (permutations) का समूह है _________ ___ __________ _______ ______ _______ ___ ______.
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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.
Find all the units of . Let R be a commutative ring with unity and
. Prove that
using the Fundamental Theorem of Homomorphism. Hence show that
.
(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)
Explicitly give the elements and structure of the group Sn / An, .
Show that if or
, then
, for
.
heck whether is a subring of the ring
or not. If it is, check whether or not it is an ideal of the ring also. If I is not a subring of the ring, then provide a subring of the ring.
जाँच कीजिए कि वलय
का एक उपवलय है या नहीं। यदि है, तो जाँच कीजिए कि वह इस वलय की एक गुणजावली भी है या नहीं। यदि I इस वलय का एक उपवलय नहीं है, तो इस वलय का एक उपवलय दीजिए।