Theorem 7.5 Let G be a group. (1) G has a unique identity element (2) Cancellation Laws. For all ...
Let G be a finite set with an associative, binary operation given by a table in which each element of the set G appears exactly once in each row and column. Prove that G is a group. How do you recognize the identity element? How do you recognize the inverse of an element? (abstract algebra)
(5 points each) Let G be a group, and let a € G. Let da: G+ G be defined by @a(g) = aga-l for all g E G. (a) Prove that Pa is an automorphism of G. (b) Let b E G. What is the image of the element ba under the automorphism ..? (c) Why does this imply that |ab| = |ba| for all elements a, b E G?
Only 2 and 3
1.) Let G be a finite G be a finite group of order 125, 1. e. 161-125 with the identity elemente. Assume that contains an element a with a 25 t e, Show that is cyclic 2. Solve the system of congruence.. 5x = 17 (mod 12) x = 13 mod 19) 3.) Let G be an abelian. group Let it be a subgroup o G. Show that alt -Ha for any a EG
Problem 4. Let G be a group. Recall that the order of an element g G is the smallest k such that gk = 1 (or 00, if such a k doesn't exist). (a) Find the order of each element of the symmetric group S (b) Let σ-(135)(24) and τ-(15)(23)(4) be permutations in S5. Find the cycle decompositions for (c) Let σ-(123456789). Compute ơ-i, σ3, σ-50, and σί006 (d) Find all numbers n such that Ss contains an element of...
Let G be a group of order 6 and let X be the set (a, b,c) E G3: abc That is, X is the set of triples of elements of G with the product of its coordinates equals the identity element of G (a) How many elements does X have? Hint: Every triple (a, b, c) in X is completely determined by the choice of a and b. Because once you choose a and b then c must be (ab)-1...
Exercise 2. Let he a group anith nentral element e. We denote the gronp lau on G simply by (91,92)gig2. Let X be a set. An action ofG on X is a a map that satisfies the following tuo conditions: c. Let G be a finite group. For each E X, consider the map (aje- fer all elements r X (b) 9-(92-2) for all 91,92 G and all r E X Show that is surjective and that, for all y...
let g=(x e R:x>1) be the set of all real numbers greater than 1. for X,Y e G, define x * y=xy - x-y +2. 1. show that the operation * is closed on G. 2. show that the associative law holds for *. 3.show that 2 is the identity element for the operation *. 4. show for each element a e G there exists an inverse a-1 e G.
1-5
theorem, state it. Define all terms, e.g., a cyclic group is generated by a single use a element. T encourage you to work together. If you find any errors, correct them and work the problem 1. Let G be the group of nonzero complex numbers under multiplication and let H-(x e G 1. (Recall that la + bil-b.) Give a geometric description of the cosets of H. Suppose K is a proper subgroup of H is a proper subgroup...
Question 2 please
Exercise 1. Define an operation on Z by a b= a - b. Determine ife is associative or commutative. Find a right identity. Is there a left identity? What about inverses? Exercise 2. Write a multiplication table for the set A = {a,b,c,d,e} such that e is an identity element, the product is defined for all elements and each element has an inverse, but the product is NOT associative. Show by example that it is not associative....