Prove by induction that it follows from Fundamental Theorem of Algebra that every f(x) ∈ C[x] can be written into a product of linear polynomials in C[x].
Abstract Algebra
Let n ≥ 2. Show that Sn is generated by each of the
following sets.
(a) S1 = {(1, 2), (1, 2, 3), (1, 2, 3, 4), ..., (1,
2, 3,..., n)}
(b) S2 = {(1, 2, 3, ..., n-1), (1, 2, 3, ..., n)}
Let φ : G1 → G2 be a group homomorphism. (abstract algebra)
(a) Suppose H is a subgroup of G1. Define φ(H) = {φ(h) | h ∈ H}.
Prove that φ(H) is a subgroup of G2.
(b) Let ker(φ) = {g ∈ G1 | φ(g) = e2}. Prove that ker(φ) is a
subgroup of G1.
(c) Prove that φ is a group isomorphism if and only if ker(φ) =
{e1} and φ(G1) = G2.
Abstract Algebra
For the group S4, let H be the subset of all permutations that fix
the element 4.
a) show this is a subgroup
b) describe an isomorphism from S3 to H