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Problem 4. Let G be a group. Recall that the order of an element g G...
Let G be a group of order 16, such that each element can be written (uniquely) in the form rasb, ...
Let G be a group of order 16, such that each element can be written (uniquely) in the form rasb, where a є {0, ,7} and b є {0,1). The elements r and s satisfy the relations: r"=1; s2 = 1; sr=r38. (The final relation means that an s can be moved past an r if we raise the r to the third power.) Let H = {1,s). Let . : G × G/H → G/H be the usual action...
2. Let G {g, g. . . , gn-le} be a cyclic group of order n,...
2. Let G {g, g. . . , gn-le} be a cyclic group of order n, H a group, and h є H. Define a function φ : G → H by φ(gi-hi for all 0 < i n-1. Show that φ is a group homomor- phism if and only if o(h) divides o(g). Warning: mind your modular arithmetic! [10]
Always give rigorous arguments I. (A) Let G be a group under * and let 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)...
(more questions will be posted today in about 6 hrs from now.) December 8, 2018 WORK ALL PROBLEMS. SHOW WORK & INDICATE REASONING \ 1.) Let σ-(13524)(2376)(4162)(3745). Express σ as a product...
(more questions will be posted today in about 6 hrs
from now.)
December 8, 2018 WORK ALL PROBLEMS. SHOW WORK & INDICATE REASONING \ 1.) Let σ-(13524)(2376)(4162)(3745). Express σ as a product of disjoint cycles Express σ as a product of 2 cycles. Determine the inverse of σ. Determine the order of ơ. Determine the orbits of ơ 2) Let ф : G H be a homomorphism from group G to group H. Show that G is. one-to-one if and...
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" |...
4. Recall that an element e in any group G is called an identity element if...
4. Recall that an element e in any group G is called an identity element if for every g € G, eg = g = ge. (a) Give a counterexample to prove that o is not an identity element in Sx. (b) Give a counterexample to prove that is not an identity element in Sx. (c) Give a counterexample to prove that is not an identity element in Sx. (a) Give a counterexample to prove that p is not an...
Let , where and let . Find the order of the element in G/K.
Let , where and let . Find the order of the element in G/K.
Let Ga finite abelian group. Prove that a)If pa primenumber divides G|, G has an element of order p b)If G2n with n odd...
Let Ga finite abelian group. Prove that a)If pa primenumber divides G|, G has an element of order p b)If G2n with n odd, G has exactly oneelement with order 2 Let Ga finite abelian group. Prove that a)If pa primenumber divides G|, G has an element of order p b)If G2n with n odd, G has exactly oneelement with order 2
4) Let G be a group and let a є G. The centralizer of a in...
4) Let G be a group and let a є G. The centralizer of a in G is defined as the set (i) Show that Ca(a) is a subgroup of G (ii) Find the centralizers of the elements r and y in the Dihedral group D4
3. a. Let H be a subgroup of a commutative group G. If every element h...
3. a. Let H be a subgroup of a commutative group G. If every element h ∈ H is a square in H (i.e., h = k 2 for some k ∈ H), and every element of G/H is a square in G/H, then every element of G is a square in G. b. Let G be a group and H a subgroup with [G : H] = 2. If g ∈ G has odd order (i.e., ord(g) is odd),...
Let G be a group of order 16, such that each element can be written (uniquely) in the form rasb, where a є {0, ,7} and b є {0,1). The elements r and s satisfy the relations: r"=1; s2 = 1; sr=r38. (The final relation means that an s can be moved past an r if we raise the r to the third power.) Let H = {1,s). Let . : G × G/H → G/H be the usual action...
2. Let G {g, g. . . , gn-le} be a cyclic group of order n, H a group, and h є H. Define a function φ : G → H by φ(gi-hi for all 0 < i n-1. Show that φ is a group homomor- phism if and only if o(h) divides o(g). Warning: mind your modular arithmetic! [10]
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)...
(more questions will be posted today in about 6 hrs
from now.)
December 8, 2018 WORK ALL PROBLEMS. SHOW WORK & INDICATE REASONING \ 1.) Let σ-(13524)(2376)(4162)(3745). Express σ as a product of disjoint cycles Express σ as a product of 2 cycles. Determine the inverse of σ. Determine the order of ơ. Determine the orbits of ơ 2) Let ф : G H be a homomorphism from group G to group H. Show that G is. one-to-one if and...
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" |...
4. Recall that an element e in any group G is called an identity element if for every g € G, eg = g = ge. (a) Give a counterexample to prove that o is not an identity element in Sx. (b) Give a counterexample to prove that is not an identity element in Sx. (c) Give a counterexample to prove that is not an identity element in Sx. (a) Give a counterexample to prove that p is not an...
4) Let G be a group and let a є G. The centralizer of a in G is defined as the set (i) Show that Ca(a) is a subgroup of G (ii) Find the centralizers of the elements r and y in the Dihedral group D4
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