### Ask Your question

Need a fast expert's response?

Submit orderand get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!

### Search & Filtering

Let p be a prime and a ∈N such that 50 a|p . Show that 50 50 p .

Which of the following statements are true? Give reasons for your answers. Marks will

only be given for valid reasons.

i) If σ is an even permutation, then I

2 σ = .

ii) If G is a group such that m (o G) = 2 , where m∈N , then G has a subgroup of order

m.

iii) If G is of order 25, then = < x >

α

x generates G , where α is a factor of 25.

iv) { , , , } α1 α2 K αn

is a set only if all the s αi

follow a given rule.

v) The characteristic of a field containing )1 50( − elements is 50.

vi) Every subring of a non-commutative ring is non-commutative.

vii) If ) ( ,R ,

1 2

∗ ∗ is a ring, then 1 2 1 2 1 ψ : R × R → R :ψ(r r, ) = r ∗ r is a binary operation.

viii) Not every polynomial that is irreducible over ]x[ Q is irreducible over ]x[ Z .

ix) If .), ( ,D + is an integral domain such that 1∈D , then D is a field.

x) The function f , defined by

3 x

x 1

)x(f

−

−

= , has the same set as domain and as

range.

only be given for valid reasons.

i) If σ is an even permutation, then I

2 σ = .

ii) If G is a group such that m (o G) = 2 , where m∈N , then G has a subgroup of order

m.

iii) If G is of order 25, then = < x >

α

x generates G , where α is a factor of 25.

iv) { , , , } α1 α2 K αn

is a set only if all the s αi

follow a given rule.

v) The characteristic of a field containing )1 50( − elements is 50.

vi) Every subring of a non-commutative ring is non-commutative.

vii) If ) ( ,R ,

1 2

∗ ∗ is a ring, then 1 2 1 2 1 ψ : R × R → R :ψ(r r, ) = r ∗ r is a binary operation.

viii) Not every polynomial that is irreducible over ]x[ Q is irreducible over ]x[ Z .

ix) If .), ( ,D + is an integral domain such that 1∈D , then D is a field.

x) The function f , defined by

3 x

x 1

)x(f

−

−

= , has the same set as domain and as

range.

Prove that R^(n)/R^(m) ~ R^(n-m)

as groups, where n, m∈ N, n ≥ m.

as groups, where n, m∈ N, n ≥ m.

Prove that if G ≠ {e} and G has no proper non-trivial subgroup, then G is finite and o(G) is a prime number.

How many Sylow 5-subgroups, Sylow 3-subgroups and Sylow 2-subgroups can a

group of order 200 have? Give reasons for your answers.

group of order 200 have? Give reasons for your answers.

Let G be a group, H∆ G and β ≤ G/H . Let A = {x ∈G | Hx∈β} . Show that

i) A ≤ G ,

ii) H∆ A ,

iii) β = A/H .

i) A ≤ G ,

ii) H∆ A ,

iii) β = A/H .

For x ∈ G , define Hx = { g-¹ xg | g ∈ G } . Under what conditions on x will Hx ≤ G ?

Further, if Hx ≤ G , will Hx ∆ G ? Give reasons for your answer.

Further, if Hx ≤ G , will Hx ∆ G ? Give reasons for your answer.

Give an example, with justification, of a group G with subgroups H and K such

that HK is not a subgroup of G .

that HK is not a subgroup of G .

Let G be a group and H ≤ G . Prove that the only right coset of H in G that is a

subgroup of G is H itself.

subgroup of G is H itself.

Write the Cayley tables for addition and multiplication in Z7.