Give the lattice of subgroups of (Z2,), of (S,,0), and of (Z(20) ,)

Let x, y be elements in a group G. Prove thatx^(−1). y^n. x = (x^(−1).yx)^nfor all n ∈ Z.

In which of the listed semigroups with identity is the element a invertible and has finite order?

Consider the elliptic-curve group defined by { (x,y) | x,y ∈ Z7 and x2 mod 7 = x3 + 2x +3 mod 7 } (ie, the group you get when a=2, b=3, and p=7). What is (2,1) + (2,1) in this group? Write your answer as an ordered pair of integers with no spaces.

Also indicate which theorem or property you are using please

Let d≥ 1 and let F be a field, and define a group of matrices G ≤ GLd+1(F) by A = {(^ ;) Prove that G is isomorphic to the group G A 1 € GL₁(F), v € Fª}. Affa(F) = {TA,v: A = GLd(F), v € Fd} considered in the previous question. (You do not need to prove that G is a group.)

Consider the elliptic-curve group defined by { (x,y) | x,y ∈ Z7 and x2 mod 7 = x3 + 2x +3 mod 7 } (ie, the group you get when a=2, b=3, and p=7). What is (2,1) + (3,1) in this group? Write your answer as an ordered pair of integers with no spaces.