If H and K are subgroups of G, IH|= 16 and |KI=28 thena possible value of |HNK| is 8. 6. 16
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Q: If H and K are subgroups of G, |H|- 16 and K-28 then a possible value of HNK| is 16
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Q: If H and K are subgroups of G, IH|= 16 and |K|=28 then a possible value of IHNK| is 16 8. Activate…
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Q: If H and K are subgroups of G, IH|= 20 and |K|=32 then a possible value of |HNK| is O 2 O 8 O 16
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Q: If H and K are subgroups of G, |H|= 16 and |K|=28 then a possible value of |HNK| is * 4 O 16
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Q: If H and K are subgroups of G, |H|= 18 and |Kl=30 then a possible value of |HNK| is O18 8. O 4
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Q: If H and K are subgroups of G, |H|= 18 and |K|=30 then a possible value of |HNK| is * 18 8 6. 4
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Q: If H and K are subgroups of G, IH|= 16 and |K|=28 then a possible value of |HNK| is * 6 4 O 16
A: solution of the given problem is below...
Q: If H and K are subgroups of G, |H|= 16 and |K|=28 then a possible value of |HNK| is * 8. O 16 4 O 6
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Q: If H and K are subgroups of G, |H|= 16 and |K|=28 then a possible value of |HNK| is 8 O 16 4 6
A: Answer is 4.
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Q: If H and K are subgroups of G. IH|- 20 and IK-32 then a possible value of HNK| is 16 8.
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Q: If H and K are subgroups of G, |H|= 18 and |K]=30 then a possible value of |HOK| is O 4 O 18 O 8
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Q: f H and K are subgroups of G, IH|= 20 and K|=32 then a possible value of |HOK[ is * O 16
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Q: If H and K are subgroups of G, |H|= 20 and |K]=32 then a possible value of IHNKI is O 2 16
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- 5. For any subgroup of the group , let denote the product as defined in Definition 4.10. Prove that corollary 4.19:Exercises 31. Let be a group with its center: . Prove that if is the only element of order in , then .11. Assume that are subgroups of the abelian group such that the sum is direct. If is a subgroup of for prove that is a direct sum.
- Consider the additive group of real numbers. Prove or disprove that each of the following mappings : is an automorphism. Equality and addition are defined on in Exercise 52 of section 3.1. a. (x,y)=(y,x) b. (x,y)=(x,y) Sec. 3.1,52 Let G1 and G2 be groups with respect to addition. Define equality and addition in the Cartesian product by G1G2 (a,b)=(a,b) if and only if a=a and b=a (a,b)+(c,d)=(ac,bd) Where indicates the addition in G1 and indicates the addition in G2. Prove that G1G2 is a group with respect to addition. Prove that G1G2 is abelian if both G1 and G2 are abelian. For notational simplicity, write (a,b)+(c,d)=(a+c,b+d) As long as it is understood that the additions in G1 and G2 may not be the same binary operations.13. Assume that are subgroups of the abelian group . Prove that if and only if is generated byLet H be a subgroup of a group G. Prove that gHg1 is a subgroup of G for any gG.We say that gHg1 is a conjugate of H and that H and gHg1 are conjugate subgroups. Prove that H is abelian, then gHg1 is abelian. Prove that if H is cyclic, then gHg1 is cyclic. Prove that H and gHg1 are isomorphic.
- Let G be an abelian group of order 2n, where n is odd. Use Lagranges Theorem to prove that G contains exactly one element of order 2.Exercises 8. Find an isomorphism from the group in Example of this section to the multiplicative group . Sec. 16. Prove that each of the following sets is a subgroup of , the general linear group of order over .Suppose that the abelian group G can be written as the direct sum G=C22C3C3, where Cn is a cyclic group of order n. Prove that G has elements of order 12 but no element of order greater than 12. Find the number of distinct elements of G that have order 12.
- Prove or disprove that H={ hGh1=h } is a subgroup of the group G if G is abelian.Let G1 and G2 be groups with respect to addition. Define equality and addition in the Cartesian product by G1G2 (a,b)=(a,b) if and only if a=a and b=a (a,b)+(c,d)=(ac,bd) Where indicates the addition in G1 and indicates the addition in G2. Prove that G1G2 is a group with respect to addition. Prove that G1G2 is abelian if both G1 and G2 are abelian. For notational simplicity, write (a,b)+(c,d)=(a+c,b+d) As long as it is understood that the additions in G1 and G2 may not be the same binary operations. (Sec. 3.4,27, Sec. 3.5,14,15,27,28, Sec. 3.6,12, Sec. 5.1,51) Sec. 3.4,27 Prove or disprove that each of the following groups with addition as defined in Exercises 52 of section 3.1 is cyclic. a. 23 b. 24 Sec. 3.5,14,15,27,28, Consider the additive group of real numbers. Prove or disprove that each of the following mappings : is an automorphism. Equality and addition are defined on in Exercise 52 of section 3.1. a. (x,y)=(y,x) b. (x,y)=(x,y) Consider the additive group of real numbers. Prove or disprove that each of the following mappings : is an isomorphism. a. (x,y)=x b. (x,y)=x+y Consider the additive groups 2, 3, and 6. Prove that 6 is isomorphic to 23. Let G1, G2, H1, and H2 be groups with respect to addition. If G1 is isomorphic to H1 and G2 is isomorphic to H2, prove that G1G2 is isomorphic to H1H2. Sec. 3.6,12 Consider the additive group of real numbers. Let be a mapping from to , where equality and addition are defined in Exercise 52 of Section 3.1. Prove or disprove that each of the following mappings is a homomorphism. If is a homomorphism, find ker , and decide whether is an epimorphism or a monomorphism. a. (x,y)=xy b. (x,y)=2x Sec. 5.1,51 Let R and S be arbitrary rings. In the Cartesian product RS of R and S, define (r,s)=(r,s) if and only if r=r and s=s (r1,s1)+(r2,s2)=(r1+r2,s1+s2), (r1,s1)(r2,s2)=(r1r2,s1s2). a. Prove that the Cartesian product is a ring with respect to these operations. It is called the direct sum of R and S and is denoted by RS. b. Prove that RS is commutative if both R and S are commutative. c. Prove that RS has a unity element if both R and S have unity elements. d. Give an example of rings R and S such that RS does not have a unity element.Prove part c of Theorem 3.4. Theorem 3.4: Properties of Group Elements Let G be a group with respect to a binary operation that is written as multiplication. The identity element e in G is unique. For each xG, the inverse x1 in G is unique. For each xG,(x1)1=x. Reverse order law: For any x and y in G, (xy)1=y1x1. Cancellation laws: If a,x, and y are in G, then either of the equations ax=ay or xa=ya implies that x=y.