Let N and H be groups, and here for a homomorphism f:
H --> Aut(N) =...
Let N and H be groups, and here for a homomorphism f:
H --> Aut(N) = group automorphism,
let N x_f H be the corresponding semi-direct product.
Let g be in Aut(N), and k be in Aut(H), Let C_g:
Aut(N) --> Aut(N) be given by
conjugation by g.
Now let z := C_g * f * k: H --> Aut(N), where *
means composition.
Show that there is an isomorphism
from Nx_f H to Nx_z H, which takes the natural...
Let S = {A, B, C, D, E, F, G, H, I, J} be the set...
Let S = {A, B, C, D, E, F, G, H, I, J} be the set consisting of
the following elements:
A = N, B = 2N , C = 2P(N) , D = [0, 1), E = ∅, F = Z × Z, G = {x
∈ N|x 2 + x < 2}, H = { 2 n 3 k |n, k ∈ N}, I = R \ Q, J =
R.
Consider the relation ∼ on S given...
) In the Extra Problem on PS 2 we defined a map φ:M2,2(Z) →
M2,2(Z2) by...
) In the Extra Problem on PS 2 we defined a map φ:M2,2(Z) →
M2,2(Z2) by the formula φ a b c d = a mod (2) b mod (2) c mod (2) d
mod (2)
On PS 2 you showed that φ is a ring homomorphism and that ker(φ)
= 2a 2b 2c 2d a, b, c, d ∈ Z We know the kernel of any ring
homomorphism is an ideal. Let I = ker(φ).
(a) (6 points) The...
1. Let V and W be finite-dimensional vector spaces over field F
with dim(V) = n...
1. Let V and W be finite-dimensional vector spaces over field F
with dim(V) = n and dim(W) = m, and
let φ : V → W be a linear transformation.
A) If m = n and ker(φ) = (0), what is im(φ)?
B) If ker(φ) = V, what is im(φ)?
C) If φ is surjective, what is im(φ)?
D) If φ is surjective, what is dim(ker(φ))?
E) If m = n and φ is surjective, what is ker(φ)?
F)...
f H and K are subgroups of a group G, let (H,K) be the subgroup
of...
f H and K are subgroups of a group G, let (H,K) be the subgroup
of G generated by the elements {hkh−1k−1∣h∈H, k∈K}.
Show that :
H◃G if and only if (H,G)<H
9. Let S = {a,b,c,d,e,f,g,h,i,j}.
a. is {{a}, {b, c}, {e, g}, {h, i, j}} a...
9. Let S = {a,b,c,d,e,f,g,h,i,j}.
a. is {{a}, {b, c}, {e, g}, {h, i, j}} a partition of S?
Explain.
b. is {{a, b}, {c, d}, {e, f}, {g, h}, {h, i, j}} a partition
of S? Explain. c. is {{a, b}, {c, d}, {e, f}, {g, h}, {i, j}} a
partition of S? Explain.