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Abstract Algebra
Let (C*, • ) denote the group of non-zero
complex numbers and
S= {z belongs to C*| IZI = 1 }. Show that
C* / S isomorphic to R+ where (R+, • ) is the group of
positive real numbers.
complex numbers and
S= {z belongs to C*| IZI = 1 }. Show that
C* / S isomorphic to R+ where (R+, • ) is the group of
positive real numbers.
Abstract Algebra
Let I be an ideal of a ring R. Define
[R:l]={x belongs to R | rx belongs to I for all r belongs to R}
Prove that
(i) [R: I] is an ideal of R.
(ii) I subset of [R : I]
[R:l]={x belongs to R | rx belongs to I for all r belongs to R}
Prove that
(i) [R: I] is an ideal of R.
(ii) I subset of [R : I]
Abstract Algebra
Let R[x] denote the set of all polynomials in x with real coefficients. On R[x], define a
relation ~ by f(x) ~g(x) if f'(x) = g'(x), where f'(x) is the derivative of f(x). Show that ~ is an equivalence relation on R[x]. For any f(x) E R[x], determine the equivalence class [f(x)].
Abstract Algebra
There is an injective ring homomorphism from M2 [Z] to M2[Z].Is it true?
Abstract Algebra
Show that R = {a + b√-3 a, b belongs to Z} i s not a
unique factorization domain by expressing
4 as a product of two irreducible elements in
R in two different ways.
unique factorization domain by expressing
4 as a product of two irreducible elements in
R in two different ways.
Abstract Algebra
Prove that Z[√-3 ] is not a U.F.D
Abstract Algebra
If G is a finite commutative group of order n
and if a prime p divides n, show that the
number of Sylow-p subgroups of G is one.
Find the unique Sylow-3 and Sylow-2
subgroups of the cyclic group Z24.
and if a prime p divides n, show that the
number of Sylow-p subgroups of G is one.
Find the unique Sylow-3 and Sylow-2
subgroups of the cyclic group Z24.
Abstract Algebra
If (G,•) is a group, then f:G×G→G, defined by f(g,h) =g², is a binary operation? Is it true?
Abstract Algebra
Is it true that if f:R→S is a ring homomorphism between two rings with unity R and S then f+1; defined by (f+1) (r) =f(r) +1 is also a ring homeomorphism from R to S?
Abstract Algebra
Assume that (G,*) is a cyclic group and that H is a subgroup of G. Explain why H is a normal subgroup G,and prove that G/H is cyclic.
