1. Let a and b be elements of a group . Prove that ab and ba...
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?
Homework Answers
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Let a and b be elements of a group G such that b has order 2...
Let a and b be elements of a group G such that b has order 2 and
ab=ba^-1
12. Let a and b be elements of a group G such that b has order 2 and ab = ba-1. (a) Show that a” b = ba-n for all integers n. Hint: Evaluate the product (bab)(bab) in two different ways to show that ba+b = a-2, and then extend this method. (b) Show that the set S = {a”, ba" |...
2. Let G be an abelian group. Suppose that a and b are elements of G...
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 H- ta + bija, b e R, ab 20). Prove or disaprove that H...
1. Let H- ta + bija, b e R, ab 20). Prove or disaprove that H is a subgroup of C under addition. 2. Let a and b be elements of an Abelian group and let n be any integer. Prove that (ab)"- a
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 an...
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.
44. a.Let A and B be two 2 × 2 matrices,Let Tr denote the trace and det denote the determinant. Prove that Tr(AB)-Tr(BA) and det(AB) - det(BA). b. If A is any matrix in SLa(R), prove that det ((-...
44. a.Let A and B be two 2 × 2 matrices,Let Tr denote the trace and det denote the determinant. Prove that Tr(AB)-Tr(BA) and det(AB) - det(BA). b. If A is any matrix in SLa(R), prove that det ((-A-t +1 where t = Tr(A).
44. a.Let A and B be two 2 × 2 matrices,Let Tr denote the trace and det denote the determinant. Prove that Tr(AB)-Tr(BA) and det(AB) - det(BA). b. If A is any matrix in SLa(R), prove...
can you answer 4 and 5 please 24. (a) Prove that if G is an abelian...
can
you answer 4 and 5 please
24. (a) Prove that if G is an abelian group, then (ab) = a*b for any a, b e G. (b) Find an example of a non-abelian group and two elements a and b such that (ab) aºb 5. (a) Prove that if G is a group and a, b e G, then (ab) -1 = b-'a-1 (b) Find an example of a group and two elements a and b such that (ab)...
Let G be a group, and let a ∈ G. Let φa : G −→ G...
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...
6.3.3 Let G be a group of order p? Prove that either G is abelian or...
6.3.3 Let G be a group of order p? Prove that either G is abelian or its center has exactly p elements.
Let a and b be elements of a group G such that b has order 2 and
ab=ba^-1
12. Let a and b be elements of a group G such that b has order 2 and ab = ba-1. (a) Show that a” b = ba-n for all integers n. Hint: Evaluate the product (bab)(bab) in two different ways to show that ba+b = a-2, and then extend this method. (b) Show that the set S = {a”, ba" |...
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 H- ta + bija, b e R, ab 20). Prove or disaprove that H is a subgroup of C under addition. 2. Let a and b be elements of an Abelian group and let n be any integer. Prove that (ab)"- a
44. a.Let A and B be two 2 × 2 matrices,Let Tr denote the trace and det denote the determinant. Prove that Tr(AB)-Tr(BA) and det(AB) - det(BA). b. If A is any matrix in SLa(R), prove that det ((-A-t +1 where t = Tr(A).
44. a.Let A and B be two 2 × 2 matrices,Let Tr denote the trace and det denote the determinant. Prove that Tr(AB)-Tr(BA) and det(AB) - det(BA). b. If A is any matrix in SLa(R), prove...
can
you answer 4 and 5 please
24. (a) Prove that if G is an abelian group, then (ab) = a*b for any a, b e G. (b) Find an example of a non-abelian group and two elements a and b such that (ab) aºb 5. (a) Prove that if G is a group and a, b e G, then (ab) -1 = b-'a-1 (b) Find an example of a group and two elements a and b such that (ab)...
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...
6.3.3 Let G be a group of order p? Prove that either G is abelian or its center has exactly p elements.
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