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

= , has the same set as domain and as
Prove that R^(n)/R^(m) ~ R^(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.
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 .
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.
Give an example, with justification, of a group G with subgroups H and K such
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.
Write the Cayley tables for addition and multiplication in Z7.
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