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.
i) False – The Euler-phi function (\phi(n)) counts numbers coprime to (n). There is no general equation given; without specifying the equation, this cannot be true in general. For example, (\phi(6) = 2) and (\phi(8) = 4); any arbitrary equation involving (\phi(n)) may fail. ii) True – If (f: G_1 \to G_2) is a group homomorphism, then (f(e_{G_1}) = e_{G_2}). This follows from the homomorphism property (f(e_{G_1}) = _____ ____ _________ __________ ________ ______ ________ _____ ___.
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समुच्चय पर विचार कीजिए। X पर को
द्वारा परिभाषित कीजिए।
के सभी मात्रकों को ज्ञात कीजिए।
समूह , के अवयवों और इसकी संरचना को स्पष्ट रूप में दीजिए।
सिद्ध कीजिए कि (n बार)
और
.
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।
Let G be a group such that is cyclic. Prove that G is abelian.
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.