Let a and b be elements of a group G such that b has order 2 and ab=ba^-1
1. Suppose a and b are elements of a group G. Prove, by induction, (bab−1)n = banb−1 . Hence prove that if a has order m, then bab−1 also has order m. Deduce from question (#1) that in any group ab and ba have the same order (you may assume ab has finite order). The assertion in Question (#1) can be generalized to an assertion about isomorphisms. State and prove it.
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...
2. Let G be an abelian group. Suppose that a and b are elements of G of finite order and that the order of a is relatively prime to the order of b. Prove that <a>n<b>= <1> and <a, b> = <ab> .
1. Let a and b be elements of a group . Prove that ab and ba have the same order. 2. Show by example that the product of elements of nite order in a group need not have nite order. What if the group is abelian?
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
please look at red line please explain why P is normal thanks Proposition 6.4. There are (up to isomorphism) exactly three di groups of order 12: the dihedral group De, the alternating group A, and a generated by elements a,b such that lal 6, b a', and ba a-b. stinct nonabelian SKETCH OF PROOF. Verify that there is a group T of order 12 as stated (Exercise 5) and that no two of Di,A,T are isomorphic (Exercise 6). If G...
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, and let a ∈ G. Let φa : G −→ G be defined by φa(g) = aga−1 for all g ∈ G. (a) Prove that φa is an automorphism of G. (b) Let b ∈ G. What is the image of the element ba under the automorphism φa? (c) Why does this imply that |ab| = |ba| for all elements a, b ∈ G? 9. (5 points each) Let G be a group, and let...
1. Let G = {a, b, c, d, e} be a set with an associative binary operation multiplication such that ab = ba = d, ed = de = c. Prove that G under this multiplication cannot consist of a group. Hint: Assume that G under this operation does consist of a group. Try to complete the multiplication table and deduce a contradiction. 2. Let G be a group containing 4 elements a, b, c, and d. Under the group...