Introduction to Modern Algebra

# 1. Let G be a group with the unit element 1. Suppose N_{1}, N_{2} are normal subgroups of G and N_{1} ∩ N_{2} = {1}. Prove that for any n_{1} ∈ N_{1}, n_{2} ∈ N_{2}, n_{1}n_{2} = n_{2}n_{1}.

# 2. Let H_{1}, H_{2} be two Sylow-5 subgroups of S_{7} (the permutation group of a set of 7 elements). Prove that H_{1}, H_{2} are conjugate i.e. there exists r ∈ S_{7} such that H_{1} = rH_{2}r^{−1}.

# 3. Let R be a commutative ring with the unit element 1, I ⊂ R and ideal, and r ∈ R. Define

I : r = {x ∈ R | xr ∈ I}.

(1) Prove that I : r is an ideal of R.

(2) Prove that the submodule R(r + I)/I ⊂ R/I is isomorphic to R/(I : r) as a module over R.

# 4. Consider ℂ^{2} as a vector space over ℝ. Define f : ℂ^{2} → ℂ^{2} by

f : (a, b) → (a + b̄, a − b̄).

Find the dimensions of the image and the kernel of this linear transformation over ℝ.

# 5. Let K be the splitting field of the polynomial x^{63} −1 over ℚ. Find the number of all the fields F ’s such that F ⊂ K and [F : ℚ] = 2.

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