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.
Let H be the subset of all skew-symmetric matrices in
M3x3
a.) prove that H is a subspace of M3x3 by checking
all three conditions in the definition of subspace.
b.) Find a basis for H. Prove that your basis is actually a
basis for H by showing it is both linearly independent and spans
H.
c.) what is the dim(H)
Let H and K be subgroups of a group G so that for all h in H and
k in K there is a k' in K with hk = k'h. Proposition 2.3.2 shows
that HK is a group. Show that K is a normal subgroup of HK.
(abstract algebra) Let F be a field. Suppose f(x), g(x), h(x) ∈
F[x]. Show that the following properties hold:
(a) If g(x)|f(x) and h(x)|g(x), then h(x)|f(x).
(b) If g(x)|f(x), then g(x)h(x)|f(x)h(x).
(c) If h(x)|f(x) and h(x)|g(x), then h(x)|f(x) ± g(x).
(d) If g(x)|f(x) and f(x)|g(x), then f(x) = kg(x) for some k ∈ F
\ {0}
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 G be an abelian group and K is a subset of G.
if K is a subgroup of G , show that G is finitely generated if
and only if both K and G/K are finitely generated.
Let G be a group acting on a set S, and let H be a group acting
on a set T. The product group G × H acts on the disjoint union S ∪
T as follows. For all g ∈ G, h ∈ H,
s ∈ S and t ∈ T,
(g, h) · s = g · s, (g, h) · t = h · t.
(a) Consider the groups G = C4, H = C5,
each acting...