Check whether or not is a subgroup of
i) ,
ii)
Consider (A={z\in\mathbb{C}^* \mid |z|\in\mathbb{Q}}). For ((\mathbb{C}^*,\cdot)): The set (A) is nonempty since (1\in A) and (|1|=1\in\mathbb{Q}). If (z,w\in A), then ________ _________ _________ ________ _____ _________.
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_____ _____ ________ ______ ____ _______ ___ _____ ____.
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एक ऐसा समूह G तथा G की एक ऐसी समाकारिता ज्ञात कीजिए, जिससे कि
हो तथा
हो। क्या G आबेली होगा? अपने उत्तर के लिए कारण दीजिए।
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.
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।
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.
जाँच कीजिए कि (X, ) एक समूह है या नहीं।
Give an example, with justification, of a commutative subgroup of a non-commutative group.
दर्शाइए कि के अनंततः अनेक मान हैं जिनके लिए
में
अखंडनीय है।