Exercise 2. Let he a group anith nentral element e. We denote the gronp lau on G simply by (91,92...
Let A be a subset of a finite group G with |A| > |G|/2. Prove that every element of G can be written as the product of two elements of A. Is this also always true when |A| = |G|/2?
Please answer all parts. Thank you!
20. Let R be a commutative ring with identity. We define a multiplicative subset of R to be a subset S such that 1 S and ab S if a, b E S. Define a relation ~ on R × S by (a, s) ~ (a, s') if there exists an s"e S such that s* (s,a-sa,) a. 0. Show that ~ is an equivalence relation on b. Let a/s denote the equivalence class...
2. Assume the group G acts on the set S. For E S, define Then G is a subgroup of G , which is called the stabilizer of r. The set is called the orbit of r (a) Consider the map ф' G S, defined by фг (g) :-9-x. Prove that there is one map (and only one) : G/G, S such that Vz ยู่'z q (where q: G -G/G, is the quotient map). (b) Prove that is injective. (Hint:...
Let G be a group and g E G such that g) is finite. Let og be the automorphism of G given by 09(x) = grg- (a) Prove that $, divides g. (b) Find an element b from a group for which 1 < 0) < 1b.
2. Assume the group G acts on the set S. For E S, define Then G is a subgroup of G , which is called the stabilizer of r. The set is called the orbit of r (a) Consider the map ф' G S, defined by фг (g) :-9-x. Prove that there is one map (and only one) : G/G, S such that Vz ยู่'z q (where q: G -G/G, is the quotient map). (b) Prove that is injective. (Hint:...
2. Let f : A ! B. DeÖne a relation R on A by xRy i§ f (x) = f
(y). a. Prove that R is an equivalence relation on A. b. Let Ex =
fy 2 A : xRyg be the equivalence class of x 2 A. DeÖne E = fEx : x
2 Ag to be the collection of all equivalence classes. Prove that
the function g : A ! E deÖned by g (x) = Ex is...
Always give rigorous arguments I. (A) Let G be a group under * and let g E G with o(g) = n (finite) (i) Show that g can never go back to any previous positive power of g* (1k< n) when taking up to the nth power (cf. g), e., that there are no integers k and m such that 1< k<m<n and such that g*-gm (ii) How many elements of the set (e, g,g2.... .g"-) are actually distinct? (iii)...
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...
Theorem 7.5 Let G be a group. (1) G has a unique identity element (2) Cancellation Laws. For all a, b,ce G, if ab ac, then b-c. For all a, b,c E G, if ba-ca, then (3) Each element of G has a unique inverse: For each a E G, there exists a unique element d e G such that ad-e and da e . Prove that each element of a finite group G appears exactly once in each row...