Let , where and let . Find the order of the element in G/K.
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 16, such that each element can be written (uniquely) in the form rasb, where a є {0, ,7} and b є {0,1). The elements r and s satisfy the relations: r"=1; s2 = 1; sr=r38. (The final relation means that an s can be moved past an r if we raise the r to the third power.) Let H = {1,s). Let . : G × G/H → G/H be the usual action...
3. a. Let H be a subgroup of a commutative group G. If every element h ∈ H is a square in H (i.e., h = k 2 for some k ∈ H), and every element of G/H is a square in G/H, then every element of G is a square in G. b. Let G be a group and H a subgroup with [G : H] = 2. If g ∈ G has odd order (i.e., ord(g) is odd),...
(1 point) Let x be an element of order 26 in a group G (not necessarily cyclic, finite, or Abelian). How many distinct subgroups of G are contained in (x)?
Let Ga finite abelian group. Prove that a)If pa primenumber divides G|, G has an element of order p b)If G2n with n odd, G has exactly oneelement with order 2 Let Ga finite abelian group. Prove that a)If pa primenumber divides G|, G has an element of order p b)If G2n with n odd, G has exactly oneelement with order 2
19. Let H and K be subgroups of a group G, where H = 9, 1K1 =12 and where the index [G:HNK] =IGI. Find (HNKI.
You are given elements g, h, k of a finite abelian group G of orders 306030, 215447282, 116699, respectively. Use g, h, k to construct an element of G of order 108954317152067.
Define σ as an element of S_z, where S_z is the collection of all permutations of all integers, by σ(k) = 3-k for all k element of all Integers. Find all the Orbits of σ.
(3) (7 points) Let G be a finite abelian group of order n. Let k be relatively prime to n. Prove the map : G G given by pla) = ak is an automor- phism of G
Let G be a finite group of order n. Let φ : G → G be the function given by φ(x) = z'n where rn E N. If gcd(rn, n) = 1, show that φ s an injective map. Let G be a finite group of order n. Let φ : G → G be the function given by φ(x) = z'n where rn E N. If gcd(rn, n) = 1, show that φ s an injective map.