Suppose φ:G→G is a group homomorphism, φ is not the trivial map, and |G|=p,where p is...
Suppose φ:G→G is a group homomorphism, φ is not the trivial map, and |G|=p,where p is a prime number. Prove that G∼=Im(φ), where Im(φ) is the image of φ.
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Suppose φ:G→G is a group homomorphism, φ is not the trivial map,
and |G|=p,where p is...
= e, then d is the trivial homomorphism 6. Suppose that ø : D7 -> G...
= e, then d is the trivial homomorphism 6. Suppose that ø : D7 -> G is a homomorphism. Prove that if ø(s) 7. For what n is there a one to one homomorphism from Z, to S7? Give an example of the homomorphism for each n.
= e, then d is the trivial homomorphism 6. Suppose that ø : D7 -> G is a homomorphism. Prove that if ø(s) 7. For what n is there a one to one...
Let φ : G → H be any group homomorphism. Prove that φ is 1-1 if...
Let φ : G → H be any group homomorphism. Prove that φ is 1-1 if and only if ker(φ) = {e}.
Let a : G + H be a homomorphism. Which of the following statements must necessarily...
Let a : G + H be a homomorphism. Which of the following statements must necessarily be true? Check ALL answers that are necessarily true. There may be more than one correct answer. A. If kera is trivial (i.e., ker a = {eg}), then a is injective. B. If the image of a equals H, then a is injective. C. The first isomorphism theorem gives an isomorphism between the image of a and a certain quotient group. D. The first...
15. If φ: Sn Sn is a group homomorphism, prove that φ(An) c An. (Hint: Use Lemma 4.7.) a 4.7. Let n 2 3. Every element...
15. If φ: Sn Sn is a group homomorphism, prove that φ(An) c An. (Hint: Use Lemma 4.7.) a 4.7. Let n 2 3. Every element of An can be written as the product of 3-cycles.
15. If φ: Sn Sn is a group homomorphism, prove that φ(An) c An. (Hint: Use Lemma 4.7.)
a 4.7. Let n 2 3. Every element of An can be written as the product of 3-cycles.
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 ...
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.
Suppose G is a group with IGI pr where p, q and r are distinct primes
(i) State Sylow's theorems. (ii) Suppose G is a group with IGI pr where p, q and r are distinct primes. Let np, nq and nr, denote, respectively, the number of Sylow p, q- and r-subgroups of G. Show that Hence prove that G is not a simple group. (iii) Prove that a group of order 980 cannot be a simple group.
2. Let p: G -G be a surjective group homomorphism (a) Show that if G is...
2. Let p: G -G be a surjective group homomorphism (a) Show that if G is abelian then G' is abelian. (b) Show that if G' is cyclic then there is a surjective homomorphism from (Z, +, 0) to G'. (Hint: use the fact that Z is generated by 1 and G' has a generator). (c) Use Part (a) and (b) to show that every cyclic group is abelian.
.. 1. (a) (10 points) Show that if 6: G + G' is a group homomorphism...
.. 1. (a) (10 points) Show that if 6: G + G' is a group homomorphism then Im(6) is a subgroup of G'. (b) (10 points) Utilize the above result to show that if 6: R → R' is a ring homomorphism then Im(6) is a subring of R'. Hint: By 1(a) it's enough to show closure under multipli- cation.
(3.) (a) Suppose that y: R S is a ring homomorphism. Please prove that (-a) =...
(3.) (a) Suppose that y: R S is a ring homomorphism. Please prove that (-a) = -f(a) for all a ER (b) Suppose R and S are rings. Define the zero function y: R S by pa) = Os for all GER. Is y a ring homomorphism? Please explain. (4.) Suppose that p is a prime number and 4: Z, Z, is defined by wa) = a.
(i) Determine whether φ defines a homomorphism. (ii) Find ker ф :-(g E G I ф(G)-e) and inn ф d(G). (ii) Draw Cayley...
(i) Determine whether φ defines a homomorphism. (ii) Find ker ф :-(g E G I ф(G)-e) and inn ф d(G). (ii) Draw Cayley diagrams of the domain and codomain, and arrange them so one can "visually see" the cosets of ker φ in G. Draw dotted lines around these cosets. (iv) Is the quotient G/ker ф a group? If so, what is it isomorphic to? Here is an example of Step (iii) for the map o: Z6 Z3, defined by...
= e, then d is the trivial homomorphism 6. Suppose that ø : D7 -> G is a homomorphism. Prove that if ø(s) 7. For what n is there a one to one homomorphism from Z, to S7? Give an example of the homomorphism for each n.
= e, then d is the trivial homomorphism 6. Suppose that ø : D7 -> G is a homomorphism. Prove that if ø(s) 7. For what n is there a one to one...
Let a : G + H be a homomorphism. Which of the following statements must necessarily be true? Check ALL answers that are necessarily true. There may be more than one correct answer. A. If kera is trivial (i.e., ker a = {eg}), then a is injective. B. If the image of a equals H, then a is injective. C. The first isomorphism theorem gives an isomorphism between the image of a and a certain quotient group. D. The first...
15. If φ: Sn Sn is a group homomorphism, prove that φ(An) c An. (Hint: Use Lemma 4.7.) a 4.7. Let n 2 3. Every element of An can be written as the product of 3-cycles.
15. If φ: Sn Sn is a group homomorphism, prove that φ(An) c An. (Hint: Use Lemma 4.7.)
a 4.7. Let n 2 3. Every element of An can be written as the product of 3-cycles.
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.
2. Let p: G -G be a surjective group homomorphism (a) Show that if G is abelian then G' is abelian. (b) Show that if G' is cyclic then there is a surjective homomorphism from (Z, +, 0) to G'. (Hint: use the fact that Z is generated by 1 and G' has a generator). (c) Use Part (a) and (b) to show that every cyclic group is abelian.
.. 1. (a) (10 points) Show that if 6: G + G' is a group homomorphism then Im(6) is a subgroup of G'. (b) (10 points) Utilize the above result to show that if 6: R → R' is a ring homomorphism then Im(6) is a subring of R'. Hint: By 1(a) it's enough to show closure under multipli- cation.
(3.) (a) Suppose that y: R S is a ring homomorphism. Please prove that (-a) = -f(a) for all a ER (b) Suppose R and S are rings. Define the zero function y: R S by pa) = Os for all GER. Is y a ring homomorphism? Please explain. (4.) Suppose that p is a prime number and 4: Z, Z, is defined by wa) = a.
(i) Determine whether φ defines a homomorphism. (ii) Find ker ф :-(g E G I ф(G)-e) and inn ф d(G). (ii) Draw Cayley diagrams of the domain and codomain, and arrange them so one can "visually see" the cosets of ker φ in G. Draw dotted lines around these cosets. (iv) Is the quotient G/ker ф a group? If so, what is it isomorphic to? Here is an example of Step (iii) for the map o: Z6 Z3, defined by...
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