Let G be a group, and let N Prove that MEN. G. Assume that G:N = m. Let zeG.
let G be a finite group, prove that for every a in G there exists a positive integer n such that an=e, the identity.
Abstract Algebra 1 a) Prove that if G is a cyclic group of prime order than G has exactly two subgroups. What are they? 1 b) Let G be a group and H a subgroup of G. Let x ∈ G. Proof that if for a, b ∈ H and ax = b then x ∈ H. (If you use any group axioms, show them)
4 (a) Let G be an abelian group with identity e and let H- gEGI8-e. Prove that H is a subgroup of G 4 (a) Let G be an abelian group with identity e and let H- gEGI8-e. Prove that H is a subgroup of G
(8) Let G be a group and let H be a subgroup of G. Prove that the right cosets of H partition G, that is, G= U Hy HYEH\G and, if y, y' E G and Hyn Hy' + 0, then Hy= Hy'.
2. Let B-[vi..... Vn] be an orthonormal basis of R". Prove that the matrix P (vilv2l...Vn) is orthogonal, that is PT P I, 2. Let B-[vi..... Vn] be an orthonormal basis of R". Prove that the matrix P (vilv2l...Vn) is orthogonal, that is PT P I,
read the example and froof and answer for question 2. Example: Prove Vn EZ with n20, 8 (3-1) Proof: Let P(n) be 8 (3*-1). [Again, using the word "be" since using an equals sign with a divisibility symbol would make no sense.] Since 320-1-0 and 8 0, P(o) is true. Next, let k eZ and k 20 and assume P(k) is true. This means 8|(32-1) so 3 xeZ such that 8x 3-1, or 3 8x+1. Then 32+)-13242 -1 -3 32-1...
Let φ : G → H be any group homomorphism. Prove that φ is 1-1 if and only if ker(φ) = {e}.
Definition. Let G be a group and let a € G. The centralizer of a is C(a) = {9 € G ag = ga}, i.e. it consists of all elements in G that commute with a. (18) (a) In the group Zui, find C(3). (b) Complete and prove the following: If G is an Abelian group and a EG, then C(a) = _. (c) Prove or disprove: In every group G, there exists a E G such that C(a) =...