Let G be a finite group. Then G is a p-group if and only if |G| is a power of p. We leouo the
Q: 1. Let a and b be elements of a group G. Prove that if a E, then C. 2. Let a and b be elements of a…
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Q: 1. Let G be a group and H a nonempty subset of G. Then H <G if ab-EH whenever a,bEH
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Q: Prove: Let G be a group and let a be a non-identity element of G. Then |a| = 2 if and only if a =…
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Q: Let H be a subgroup of a group G and a, be G. Then bE aH if and only if * O a-1b eH O ab-1 eH O None…
A: We know that b∈bH (1) We know that aH = bH if and only if a-1b ∈H…
Q: Suppose that G is a cyclic group such that Ord(G) = 48. The number of subgroups that G has is * O 8…
A: Q1. Third option is correct. Q2. Second option is correct.
Q: Let G a finite group, g E G such
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Q: 9) Let H be a subgroup of a group G and a, be G. Then a e bH if and only if O b-la e H O ba e H O…
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Q: For any subset H of a group G, if ab^-1 is in H for all a,b element of H then H is a subgroup of G.…
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Q: If G = is a distant group and f: G → G is an automorph, show that f(a) is a generator of G, i.e. G =
A: We need to show every element of G is some power of f(a)
Q: Let G be a group. Show that Z(G) = nEC(a). [This means the intersection of all subgroups of the form…
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Q: If G is a finite group with |G|<160 and G has subgroups of orders 1O, 16 and 20 then the order of G…
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Q: Let G be any group and p: G -G, (g) =g1 Vg eG. Then: (a) o is a homomorphism (b) o is a homomorphism…
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Q: 1. Let G.*) be a group and a EG Suppose that a*a = a Prove or disprove that a must be the identity…
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Q: Let G be a finite group and H1, H2,…., Hk be subgroups of G. .... (a) Show that N H; = Hị n H2 n..n…
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Q: Let G be an abelian group and suppose that H and K are subgroups of G. Show that the following set…
A: According to the given information, Let G be an abelian group and suppose that H and K are subgroups…
Q: 9) Let H be a subgroup of a group G and a, bE G. Then a E bH if and only if* O ba e H O b-1a e H…
A: We will use definition of left coset
Q: Let G be a group and a ∈ G. The centralizer of a in G is equal to the centralizer of a^-1 in G.…
A: First let us see the definition of centralizer or normalizer of a in G (definition from I.N.…
Q: Let G be a group with |G|=187 then every proper subgroup of G is:
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Q: Let G be a finite group and a€G s.t |a|=12.if H= find all other generators of H.
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Q: If G is a finite group with |G|<120 and G has subgroups of orders 1O, 15 and 20 then the order of G…
A: Using Lagrange's theorem it can be written that if G is a finite group and B is a subgroup of G,…
Q: Let G be a group and a e G. Show that o(a) = o(a-). order n, then ba also has order n.
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Q: Let H be a subgroup of a group G and a, b EG. Then b E aH if and only if O None of these O ab EH О…
A: The solution is :
Q: Let K and H be subgroups of a finite group G with KCHCG. If [G:H] = 4 and [H:K] = 3. Then, [G:K] =…
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Q: Let a and b belong to a group G. Find an x in G such that xabx-1 = ba.
A: a and b belong to a group G. We need to find an X in such that xabx-1 = ba.
Q: Let G be a group and a be an element of this group : then necessarily O laisIGI lal2/G] O lal=IG]
A: Given , Let G be a group and a be an element of this group
Q: (3) For any group G, Z(G) ≤ [G, G].
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Q: For any group G then G/Z(G) is abelin Select one: O True O False
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Q: If G is a finite group with |G|<180 and G has subgroups of orders 10, 18 and 30 then the order of G…
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Q: Let K and H be subgroups of a finite group G with KCHCG.If[G:H] = 4 and [H:K] = 3. Then, [G:K] = 3 4…
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Q: {hk | h ∈ H, k ∈ K}}
A: We have to prove that {hk|h∈H, k∈K} is a subgroup of G.
Q: 3) Let G be a group. Show that if Aut (G) is cyclic, then G is abelian.
A: Solution is given below
Q: 1. Let (G. *) be a group and a E G. Suppose that a *a = a · Prove or disprove that a must be the…
A: We have to solve given problem:
Q: Let G be a group and a be an element of this group then necessarily: * O lal=|G| |a|</G| O lals|G] O…
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Q: Let G be a group which is an algebraic structure.Let H and K be the subgroups of G with H⊂K⊂G. Let…
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Q: 8. Let (G,*) be a group, and let H, K be subgroups of G. Define H*K={h*k: he H, ke K}. Show that H*…
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Q: 5. Let (G, *) be a cyclic group, namely G=(a). Prove that (a). if G is finite of order n, then G…
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Q: Suppose that G is a cyclic group such that Ord(G) = 54. The number of subgroups that G has is * 10…
A: If G is cyclic group and order of G is 'n'. Then number of subgroups of G is equal to number of…
Q: If G is a finite group with |G|<180 and G has subgroups of orders 10, 18 and 30 then the order of G…
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Q: 9) Let H be a subgroup of a group G and a, bE G. Then a E bH if and only if * ba-1 E H ba E H O b-1a…
A: Q9. Third option is correct.
Q: If G is a finite group and a ∈ G, prove that |a| is finite. Hint: You just need to show that there…
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Q: If G is a finite group with IG|<160 and G has subgroups of orders 10, 16 and 20 then the order of G…
A: Given :- order of a group G... |G| < 160 Also order of subgroups of group G are 10, 16 , 20. We…
Q: 5. Let G be a group and n e Z+ be fixed. Show that H = {a" | a € G} is a subgroup of G
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Q: dicfin et Prove that a group G has exactly 3 6. - subgroups iff G is a ylic grop ef ender på pis…
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Q: Let H be a subgroup of a group G and a, be G. Then b E aH if and only if ab-1 e H O ab e H O None of…
A: Ans is given below
Q: 4. Let G be a group and let H, K be subgroups of G such that |H| = 12 and |K| = 5. Prove that HNK =…
A: We have to prove given result:
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
A: It is given that, G is a group and g∈G. To sow that if H is a sylow p-subgroup of G, then so is…
Q: a) Show that given a finite group G and g ∈ G, the subgroup generated by g is itself a group. (b)…
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Q: 9) Let H be a subgroup of a group G and a, bE G. Then a E bH if and only if * None of these b-1a e H…
A: Second option is correct.
Q: If a subgroup H of a group G is cyclic, then G must be cyclic. Select one: O True O False
A: we will give the counter example in support of our answer.
Q: 1. Let G be a group and let H, H, .. H, be the subgroups of G. The ...
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- True or false Label each of the following statements as either true or false, where H is subgroup of a group G. An element x in H has an inverse x1 in H that may be different than its inverse in G.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.True or False Label each of the following statements as either true or false. Let H be a subgroup of a finite group G. The index of H in G must divide the order of G.
- True or False Label the following statements as either true or false. 1. Every finite group of order is isomorphic to a subgroup of order of the group of all permutations on .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.True or False Label each of the following statements as either true or false. Let H be a subgroup of a group G. If hH=Hh for all hH, then H is normal in G.
- True or False Label each of the following statements as either true or false. 9. The symmetric group on elements has order .Exercises In Section 3.3, the centralizer of an element a in the group G was shown to be the subgroup given by Ca=xGax=xa. Use the multiplication table constructed in Exercise 20 to find the centralizer Ca for each element a of the octic group D4. Construct a multiplication table for the octic group D4 described in Example 12 of this section.Exercises 3. Find the order of each element of the group in Example of section. Example 3. We shall take and obtain an explicit example of . In order to define an element of , we need to specify , , and . There are three possible choices for . Since is to be bijective, there are two choices for after has been designated, and then only one choice for . Hence there are different mappings in .
- True or false Label each of the following statements as either true or false, where is subgroup of a group. 2. The identity element in a subgroup of a groupmust be the same as the identity element in.27. a. Show that a cyclic group of order has a cyclic group of order as a homomorphic image. b. Show that a cyclic group of order has a cyclic group of order as a homomorphic image.True or False Label each of the following statements as either true or false. Let H be any subgroup of a group G and aG. Then aH=Ha implies ah=ha for all h in H.