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.
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?
a) Show that the positive rationals form a group under the operation a ∗ b = ½ ab. b) Let G be a group, and suppose the elements g and h of G don’t commute. (i) Show that g doesn’t commute with gh. (ii) Find another element of G, distinct from h and gh, that doesn’t commute with g . c) Let G be a group, and let g be in G. (i) Prove that g and g^−1 have the...
Let a and b be elements of a group G. Show that if ab has finite order n, then ba also has order n.
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
Let u, b and c be elements of a group G. Prove the following:-Ord(a)=ord(bab^-1)-the order ab is the same as the order of ba.
Let the dihedral group Dn be given by the elements a of order n and b of order 2, where ba=ba-1. Find the smallest subgroup ofDn that contains a2 and b. (hint: consider two cases, depending on whether n is odd or even.)
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...
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...
(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?
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