Let a be an element of a group G such that Ord(a) = 32. If H is a normal subgroup of G, then Ord(aH) could be O None of the choices O 4 8
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Q: Let a be an element of a group G such that Ord(a) = 30. If H is a normal subgroup of G, then Ord(aH)…
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Q: 6. If G is a group and H is a subgroup of index 2 in G; then prove that H is a normal subgroup of G:
A: I have proved the definition of normal subgroup
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Q: Let H be a subgroup of a group G and a, b E G. Then be aH if and only if *
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A: Thanks for the question :)And your upvote will be really appreciable ;)
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Q: Give an example of a finite group G with two normal subgroups H and K such that G/H = G/K but H 7 K.
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Q: Let G be a group of order 24. If H is a subgroup of G, what are all the possible orders of H?
A: Given, o(G)=24 wherre H is a subgroup of G from lagrange's theoram: for any finite order group of G…
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Q: 7. Let G be a group, and let g E G. Define the centralizer, Z(g), of g in G to be the subset Z(g) =…
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Q: Let G be a group and g E G. Prove that if H is a Sylow p-group of G, then so is gHg-1
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Q: Let a be an element of a group G such that Ord(a) = 32. If H is a normal subgroup of G, then Ord(aH)…
A: Result: Let G be a group and H be a normal subgroup of G. Let 'a' be an element of G such that order…
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Q: Let a be an element of a group G such that Ord(a) = 30. If H is a normal subgroup of G, then Ord(aH)…
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Q: Prove that if H is normal in G and ø is onto, then ø[H] is normal in G'.
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Q: 6. Let G be a group of order p², where p is a prime. Show that G must have a subgroup of order p.
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- For each of the following subgroups H of the addition groups Z18, find the distinct left cosets of H in Z18, partition Z18 into left cosets of H, and state the index [ Z18:H ] of H in Z18. H= [ 8 ] .15. Assume that can be written as the direct sum , where is a cyclic group of order . Prove that has elements of order but no elements of order greater than Find the number of distinct elements of that have order .9. Consider the octic group of Example 3. Find a subgroup of that has order and is a normal subgroup of . Find a subgroup of that has order and is not a normal subgroup of .
- Exercises 13. For each of the following values of, find all subgroups of the group described in Exercise, addition and state their order. a. b. c. d. e. f.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.3. Consider the group under addition. List all the elements of the subgroup, and state its order.
- Let G be the group and H the subgroup given in each of the following exercises of Section 4.4. In each case, is H normal in G? Exercise 3 b. Exercise 4 c. Exercise 5 d. Exercise 6 e. Exercise 7 f. Exercise 8 Section 4.4 Let H be the subgroup e, of the octic group D4. Find the distinct left cosets of H in D4, write out their elements, partition D4 into left cosets of H, and give [D4:H]. Find the distinct right cosets of H in D4, write out their elements, and partition D4 into right cosets of H. Let H be the subgroup e, of the octic group D4. Find the distinct left cosets of H in D4, write out their elements, partition D4 into left cosets of H, and give [D4:H]. Find the distinct right cosets of H in D4, write out their elements, and partition D4 into right cosets of H. Let H be the subgroup e, of the octic group D4. Find the distinct left cosets of H in D4, write out their elements, partition D4 into left cosets of H, and give [D4:H]. Find the distinct right cosets of H in D4, write out their elements, and partition D4 into right cosets of H. Let H be the subgroup (1),(2,3) of S3. Find the distinct left cosets of H in S3, write out their elements, partition S3 into left cosets of H, and give [S3:H]. Find the distinct right cosets of H in S3, write out their elements, and partition S3 into right cosets of H. In Exercises 7 and 8, let G be the multiplicative group of permutation matrices I3,P3,P32,P1,P4,P2 in Example 6 of Section 3.5 Let H be the subgroup of G given by H=I3,P4={ (100010001),(001010100) }. Find the distinct left cosets of H in G, write out their elements, partition G into left cosets of H, and give [G:H]. Find the distinct right cosets of H in G, write out their elements, and partition G into right cosets of H. Let H be the subgroup of G given by H=I3,P3,P32={ (100010001),(010001100),(001100010) }. Find the distinct left cosets of H in G, write out their elements, partition G into left cosets of H, and give [G:H]. Find the distinct right cosets of H in G, write out their elements, and partition G into right cosets of H.Find the right regular representation of G as defined Exercise 11 for each of the following groups. a. G={ 1,i,1,i } from Example 1. b. The octic group D4={ e,,2,3,,,, }.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.
- a. Let G={ [ a ][ a ][ 0 ] }n. Show that G is a group with respect to multiplication in n if and only if n is a prime. State the order of G. This group is called the group of units in n and is designated by Un. (Sec 3.35, Sec 3.411, Sec 3.519) b. Construct a multiplication table for the group U7 of all nonzero elements in 7, and identify the inverse of each element. (Sec 4.4,1,19,26) Sec 3.35 5. Exercise 33 of section 3.1 shows that U1313 is a group under multiplication. List the elements of the subgroup [ 4 ] of U13, and state its order. List the elements of the subgroup [ 8 ] of U13, and state its order. Sec 3.411 11. If n is a prime, the nonzero elements of n form a group Un with respect to multiplication. For each of the following values of n, show that this group Un is cyclic. n=7 b. n=5 c. n=11 d. n=13 e. n=17 f. n=19 Sec 3.519 19. If n is a prime, Un, the set of nonzero elements of n, forms a group with respect to multiplication. Prove or disprove that the mapping :UnUn defined by the rule in Exercise 18 is an automorphism of Un. Construct a multiplication table for the group U7 of all nonzero elements in 7, and identify the inverse of each element. (Sec 4.4,1,19,26) Sec 4.4,1 1. Consider U13, the groups of units in 13 under multiplication. For each of the following subgroups H in U13, partition U13 into left cosets of H, and state the index [ U13:H ] of H in U13 H= [ 4 ] b. H= [ 8 ] Sec 4.4,19 19. Find the order of each of the following elements in the multiplicative group of units Up. [ 2 ] for p=13 b. [ 5 ] for p=13 c. [ 3 ] for p=17 d. [ 8 ] for p=17 Sec 4.4,26 26. Let p be prime and G the multiplicative group of units Up={ [ a ]p[ a ][ 0 ] }. Use Langranges Theorem in G to prove Fermats Little Theorem in the form [ a ]p=[ a ] for any a.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.