Let be a finite abelian group and
. Prove that
.
Let ( (G, \cdot) ) be a finite abelian group and ( m \in \mathbb{N} ). Define [ S = { g \in G \mid \gcd(o(g), m) = 1 }. ] We want to show that ( S \leq G ), i.e., ( S ) is a subgroup of ( G ). Since ( G ) is abelian, _______ _________ ___ __________ ____ ____ ________ ___ __________ ______ _________ _______.
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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.
Show that there are infinitely many values of for which
is irreducible in
.
मान लीजिए कि G एक ऐसा समूह है कि चक्रीय है। सिद्ध कीजिए कि G आबेली होगा।
जाँच कीजिए कि निम्नलिखित का एक उपसमूह है या नहीं :
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.
र्शाइए कि यदि या
, तो
, जहां
.
Show that M is an ideal of R.