Let G be a group, and assume that a and b are two elements of order 2 in G. If ab = ba, then what can you say about (a, b)?
Q: Consider the group (Z,*) defined as a*b=a=b , then identity (Neutral) element is
A: Given that ℤ,* is a group. where * is defined as a*b=a=b. That is a-b=0. To find the neutral element…
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: Let G be a group of order pm where p is a prime number and m is a positive integer. Show that G…
A: Let G be a group of order pm, where p is a prime number and m is a positive integer. Then,…
Q: Let G be a group and a be an element of this group then necessarily: * Jal<|G| O lal2|G| O lal=|G|
A: In the given question we have to chose the correct option from the given options of the given…
Q: The following is a Cayley table for a group G, 2 * 3 * 4 = 3 1 2. 4 主 3. 4 2 1 21 4 345
A: For group, 2*3*4=(2*3)*4.
Q: Let G be a group and a E G be a certain fixed element of G. The centralizer of a in G is C(a) = {g €…
A: Hey, since there are multiple questions posted, we will answer the first question. If you want any…
Q: If a, b are elements in a group G, show that (ab)-1= b-1a-1
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Q: Let (G,*) be an a belian group, if (H,) and (K,*) are subgroup of (G,*) then (H * K,*) is a subgroup…
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Q: Let G be a group and let a,b element of G such that (a^3)b = ba. If |a| = 4 and |b| = 2, what is…
A: see below the answer
Q: Let G be a cyclic group ; G=, then (c*b)^=c4* b4 for all a, c, b EG.
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Q: If a is an element of order 8 of a group G, and = ,then one of the following is a possible value of…
A: Given that a is an element of order 8 and a4=ak
Q: Let G be a group and a be an element of this group such that a^6=e. The possible orders of a are: *…
A: First option is correct.
Q: a) List all the subgroups of Z, e Zz. b) Is the groups Z, ® Zz and Z, isomorphic? (why?)
A: We use the fact that for distinct prime p and q Zp x Zq is isomorphic to Zpq.
Q: If H is a Sylow p-subgroup of a group, prove that N(N(H)) = N(H).
A: Let G be a finite group and H be the subset of G. Then, normalizer of H in G, when we conjugate H…
Q: If a is an element of order 8 of a group G, and
A: Let G be a group. Let a is an element of order 8 of group G. That is, a8=e where e is an…
Q: Show that each of the following is not a group. 1. * defined on Z by a*b = |a+b|
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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 *
A: So, a, b belongs to H, and we have b∈aH Hence, b = ah -- for some element of H Hence, a-1…
Q: Let G be a group and a be an element of this group such that a^6=e. The possible orders of a are: *…
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Q: Consider the group (Z,*) defined as a*b=a+b , then identity (Neutral) element is a 1 b -1…
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Q: In a group G,let a,b and ab have order 2.show that ab=ba
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Q: Let S, be the symmetric group and let a be an element of S, defined by: 1 2 3 4 5 67 8 9 ) B = (7…
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Q: Let G be a group, and let a E G. Prove that C(a) = C(a-1).
A: Given: Let G be a group and let a∈G. then we will prove C(a)=C(a-1) If C(a) be the centralizer of a…
Q: Let a and b be elements of a group. If |a| and |b| are relatively prime, show that intersects =…
A: Let m and n be the order of the elements a and b of a group G. Given that the orders of a and b are…
Q: Let (G,*) be an a belian group, if (H,*) and (K,*) are subgroup of (G,*) then (H * K,*) is a…
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Q: Let G be a group and let a be a non-identity element of G. Then |a| = 2 if and only if a = a-1.
A: Let G be a group with respect to * . Let e be the identity element,and a is non identity element.…
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: f H and K are two subgroups of a group G, then show that for any a, b ∈ G, either Ha ∩ Kb = ∅ or Ha…
A: If H and K are two subgroups of a group G, then show that for any a, b ∈ G,either Ha ∩ Kb = ∅ or Ha…
Q: If G is a group and g E G, show that the number of conjugates of g E G is [G : CG(g)]
A: Given G be a group and g∈G be an element. Let Bg be the set of all conjugate elements of g∈G.…
Q: Prove that in a group, (ab)^2=a^2b^2 if and only if ab=ba.
A: Proof:Let a,b ∈ G.Assume (ab)2 = a2b2 and that prove ab = ba as follows.
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: Given the groups R∗ and Z, let G = R∗ ×Z. Define a binary operation ◦ on G by (a, m) ◦ (b, n) = (ab,…
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Q: Let E = Q(√2, √5). What is the order of the group Gal(E/Q)?What is the order of Gal(Q(√10)/Q)?
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Q: Let a,b be elements of a group G .Assume that a has order 5 and a^3b=ba^3. prove that ab=ba.
A: Let a,b be elements of a group G .Assume that a has order 5 and a^3b=ba^3. prove that ab=ba.
Q: Prove that if (ab)' = a*b² in a group G, then ab = ba.
A: Given,ab2=a2b2To prove: ab=ba
Q: If A is a group and B is a subgroup of A. Prove that the right cosets of B partitions A
A: Given : A be any group and B be any subgroup of A. To prove : The right cosets of B partitions A.
Q: Let G be a group and a be an element of this group such that a^63e. The possible orders of a are: *…
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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: 6. If G is a group and a is an element of G, show that C(a) = C(a')
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Q: If a is an element of order 8 of a group G, and 4 = ,then one of the following is a possible value…
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Q: Let a and b be elements in a group G. Prove that ab^(n)a^(−1) = (aba^(−1))^n for n ∈ Z.
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Q: Let G be a group and a be an element of this group such that a^6=e. The possible orders of a are: O…
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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: Let (G, ) be a group. Define a new binary operation * on G by the formula a * b = b · a for all a, b…
A: We proved (G,*) is a group if it satisfied the following axioms.
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: let G be a group, a,b E G such that bab^-1 =a^r , for some r E N, where N are the natural ones,…
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Q: Compute the left regular representation of the group G = {e, a, b} given by the group table below by…
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Q: Let a,b be elements of S6 (symmetric group) where a=(1,2)(4,5) and b=(1,6,5,3,2).verify that…
A: Given: Let a,b be elements of S6 (Symmetric group) where a=(1,2)(4,5) and b=(1,6,5,3,2).
Q: Let a, b be elements of a group G. Assume that a has order 5 and a³b = ba³. Prove that ab = ba.
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Q: : Show that in a group G, if a? = e,Vx E G, then G is a commutative. %3D
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- 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 .Exercises 31. Let be a group with its center: . Prove that if is the only element of order in , then .Suppose ab=ca implies b=c for all elements a,b, and c in a group G. Prove that G is abelian.