(1) Let d and m be positive integers. (a) Prove that mZ is a subgroup of...
(2) (a) Prove that the set G = {+1, £i} is a finite subgroup of the multi- plicative group CX of nonzero complex numbers, and that the set H = {E1} is a finite subgroup of {+1, £i}. (b) Compute the index of H in G. (c) Compute the set of left cosets G/H.
27. (a) Let m and n be integers > 1 which are relatively prime. Show that the map f : Z → Z/mZ × Z/nZ whith f(x) = (x + mZ, x + nZ) is surjective (b) Prove the Chinese Remainder Theorem: If m and n are relatively prime integers > 1 and if a and b are any integers, then there exists a E Z such that b(mod n). a(mod m) and a a Hint: (a)] 27. (a) Let...
Answer Question 5 . Name: 1. Prove that if N is a subgroup of index 2 in a group G, then N is normal in G 2. Let N < SI consists of all those permutations ơ such that o(4)-4. Is N nonnal in sa? 3. Let G be a finite group and H a subgroup of G of order . If H is the only subgroup of G of order n, then is normal in G 4. Let G...
= (3) Consider the transposition Ti (2,3) in the symmetric group S3. (a) Prove that H = {e, Ti} is a subgroup of S3. (b) Compute the index of H in S3. (c) Compute the set of left cosets S3/H. (d) Compute the set of rightcosets H\S3.
Let and , the subgroup of fourth roots of unity. (a) Characterize all left cosets of H. (b) Prove or disprove that G/H isomorphic to G. G =C H={+1, i)
Let q be a prime and let m and n be non-zero integers. Prove that if m and n are coprime and q? divides mn, then q? divides m or q? divides n
4. Let H be a subgroup of a group G and let a, b e H. Using the definition of cosets, prove that Ha= Hb if and only if ab-EH.
1) Let n and m be positive integers. Prove: If nm is not divisible by an integer k, then neither n norm is divisible by k. Prove by proving the contrapositive of the statement. Contrapositive of the statement:_ Proof: Direct proof of the contrapositive
5. (FP1.73) Let a and b be positive integers such that a2b and a is even. Then 8 divides a We will prove this in steps. (a) Come up with an example of positive integers a and b such that a2 (b) Now prove the statement. You may use the work from class, but your proof should be and a is even written out as paragraphs and displayed math that flows. It should not be numbered substeps. In other words,...
Q. 5 5. Let H G be a subgroup and suppose that H,g2H.....gH are the distinet left cosets of H in G. Prove that gH - Hg for all g e G if and only if g.H Hg,, for all 2 sisr