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) =...
the following questions are relative,please solve them,
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4. Let G be a group. An isomorphism : G G is called an automorphism of G. (a) Prove that the set, Aut(G), of all automorphisms of G forms a group under composition. (b) Let g E G. Show that the map ф9: G-+ G given by c%(z)-gZg", įs an automorphism. These are called the inner automorphisms of G (c) Show that the set of all g E G such that Og-Pe...
10. Let G = D. be the dihedral group on the octagon and let N = (r) be the subgroup of G generated by r4. (a) Prove that N is a normal subgroup of G. (b) If G =D3/N, find G. (c) Using the bar notation for cosets, show that G = {e, F, 2, 3, 5, 87, 82, 83}. Hint: Show that the RHS consists of distinct elements and then use part (b). (d) Prove that G-D4. Hint: It...
Let D4 be dihedral group order 8. So D4={e, a, a^2, a^3, b, ab, a^2b, a^3b}, a^4 = e, b^2= e, ab=ba^3; A. FIND ALL THE COSETS OF THE SUBGROUP H= , list their elements. B. What is the index [D4 : H] C. DETERMINE IF H IS NORMAL
Question 4
Exercise 1. Let G be a group such that |G| is even. Show that there exists an EG,17e with x = e. Exercise 2. Let G be a group and H a subgroup of G. Define a set K by K = {z € G war- € H for all a € H}. Show that (i) K <G (ii) H <K Exercise 3. Let S be the set R\ {0,1}. Define functions from S to S by e(z)...
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)...
abstract algebra
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3. Let H be a subgroup of G with |G|/\H = 2. Prove that H is normal in G. Hint: Let G. If Hthen explain why xH is the set of all elements in G not in H. Is the same true for H.C? Remark: The above problem shows that A, is a normal subgroup of the symmetric group S, since S/A, 1 = 2. It also shows that the subgroup Rot of all rotations...
. (15 points) Let G be a group and A be a nonempty subset of G. Consider the set Co(A) = {9 € G gag- = a for all a € A}. (a) Compute Cs, ({€, (123), (132)}), where e is the identity permutation. (b) Show that CG(A) is a subgroup of G. (c) Let H be a subgroup of G. Show that H is a subgroup of Ca(H) if and only if H is abelian.
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...
7. Let G be a group and let H be a subgroup of G. Prove that the relationon G given by ab if ab-i є H is an equivalence relation.