Direct product of groups: Let (G, ∗G) and (H,
∗H) be groups, with identity elements eG and
eH, respectively. Let g be any element of G, and h any
element of H. (a) Show that the set G × H has a natural group
structure under the operation (∗G, ∗H). What
is the identity element of G × H with this structure? What is the
inverse of the element (g, h) ∈ G × H? (b) Show that the map...
Let G, H be groups and define the relation ∼= where G ∼= H if
there is an isomorphism ϕ : G → H.
(i) Show that the relation ∼= is an equivalence relation on the
set of all groups.
(ii) Give an example of two different groups that are
related.
Example 3.5: Again let X = Y = R. Define g by g(x) = x2. The
graph of this function has the familiar parabolic shape as in
Figure 3.1(b). Then for example, g([0, 1]) = [0, 1], g([1,
2]) = [1, 4], g({−1, 1}) = {1}, g−1([0, 1]) = [−1, 1], g−1([1, 2])
= [− √ 2, −1]∪[1, √ 2], g−1([0, ∞)) = R.
*I need help understanding why each example in bold is
the answer it is*
*Please explain...
Let A and B be groups, and consider the product group G=A x
B.
(a) Prove that N={(ea,b) E A x B| b E B} is a
subgroup.
(b) Prove that N is isomorphic to B
(c) Prove that N is a normal subgroup of G
(d) Prove that G|N is isomorphic to A
Let G be an abelian group.
(a) If H = {x ∈ G| |x| is odd}, prove that H is a subgroup of G.
(b) If K = {x ∈ G| |x| = 1 or is even}, must K be a subgroup of G?
(Give a proof or counterexample.)
5. Let X, Y and Z be sets. Let f : X ! Y and g : Y ! Z
functions.
(a) (3 Pts.) Show that if g f is an injective function, then f
is an injective function.
(b) (2 Pts.) Find examples of sets X, Y and Z and functions f
: X ! Y and g : Y ! Z such that g f is
injective but g is not injective.
(c) (3 Pts.) Show that if...
Let H, K be groups and α : K → Aut(H) be a homomorphism of
groups. Show that H oα K is the internal semidirect product of
subgroups which are isomorphic to H and K, respectively
Let G = Z x Z and H = {(a, b) in Z x Z | 8 divides a+b}
a. Prove directly that H is a normal subgroup in G (use the fact
that closed under composition and inverses)
b. Prove that G/H is isomorphic to Z8.
c. What is the index of [G : H]?