Given the groups R∗ and Z, let G = R∗ ×Z. Define a binary operation ◦ on G by (a, m) ◦ (b, n) = (ab, m + n). Show that G is a group under this operation.

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: Label the following statement as either true or false.   The alternating group A4 on 4 elements is…

A: We have to state whether the given statement is true or false : The given statement is : The…

Q: Show that if H and K are subgroups of an abelian group G, then {hk|h € H and k e K} is a subgroup of…

A: A set G is called a group if it satisfies four properties  Closure property:  ab∈G  where a,b ∈G…

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: Verify that (Z, O) is an infinite group, where Z is the set of integers and the binary operator O is…

A: According to the given information, it is required to verify that:

Q: Use the fact that a group with order 15 must be cyclic to prove: if a group G has order 60, then the…

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Q: Let G be a group and let a, b E G. (a) Prove that o(ab) = o(ba). (Note that we are not assuming that…

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Q: If a is a group element, prove that every element in cl(a) has thesame order as a.

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Q: For any elements a and b from a group and any integer n, prove that (a21ba)n 5 a21bna.

A: For any elements a and b form a group and any integer n, We have to prove that a-1ban=a-1bna.

Q: If G is a group and a1, a2,…, an shows that a1 * a2 *… * an is unique, regardless of the order in…

A: We are given that G is a group. (G satisfies all the group axioms). Suppose * is the defined binary…

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: Verify that (Z, O) is an infinite group, where Z is the set of integers and the binary operator O is…

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Q: Let G be a group and let r, y e G such that ya = r-ly. Use the Principle of Mathematical Induction…

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Q: Given that A and B is a group. Find out if : A→B is a homomorphism. If it is a homomorphism, also…

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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: Show that a group of order 4 either is cyclic or is isomorphic to the Klein four group {e, a, b, a…

A: Given :              A group of order 4  Corollary :                    The order of an element 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 H  and K  be subgroups of a group G with operation * .  Prove that HK  .is closed  under the…

A: Given information: H  and K  be subgroups of a group G with operation * To prove that HK is a closed…

Q: Prove that in a group, (a-1)-1 = a for all a.

A: By definition (a-1)-1=a are both elements of a-1. Since in a group each element has a unique…

Q: If H and K are subgroups of a group G, prove that ANB is a subgroup of G.

A: GIVEN if H and K are the subgroup of a G, prove that  A∩B  is a subgroup of G

Q: Let G be a group, H4G, and K < G. Prove that HK is a subgroup of G. Bonus: If in addition K 4G,…

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Q: Let C be a group with |C| = 44. Prove that Cmust contain an element of order 2.

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Q: If (G, ) is a group with a = a for all a in G then G is %D abelian

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Q: Show if the shown group is cyclic or not. If cyclic, provide its generator/s for H H = ({3* : k E…

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Q: If (G, * ) is a group with a a for all a in G then G is abelian

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Q: Let G = Z, be the cyclic group of order n, and let S c Z, \ {0}, such that S = -S, \S| = 3 and (S) =…

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Q: Let H and K be subgroups of a group G. If |H| = 63 and |K| = 45,prove that H ⋂ K is Abelian.

A: Given: The H and K are subgroups of a group G. If |H| = 63 and |K| = 45 To prove that H ⋂ K is…

Q: Show if the shown group is cyclic or not. If cyclic, provide its generator/s for H H = ({a +bv2 : a,…

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Q: If G is a group with identity e and a2 = e for all a ∈ G, then prove that G is abelian.

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Q: Show that if p and q are distinct primes, then the group ℤp × ℤq is isomorphic to the cyclic group…

A: We have to show that if p and q are distinct primes, then the group Zp×Zq is isomorphic to the…

Q: Is the set Z a group under the operation a * b = a – b + ab? Justify your answer.

A: Check the associative property. Take a = 2, b = 3 and c =4. (a*b)*c = (2*3)*4              =…

Q: any two elements x and y in a group G, there is a unique element z in G such that y = xz.

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Q: Define * on Q by a +b= qb Is Q a group under *? 210

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Q: If G is a cyclic group of order n, prove that for every element a in G,an = e.

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Q: Suppose the o and y are isomorphisms of some group G to the same group. Prove that H = {g E G| $(g)…

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Q: Let (G, ) be a group. Define a new binary operation * on G by the formula a * b = b - a for all a…

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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

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 an abelian group, then (acba)(abc)¯1 is

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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: Let c and of d be elements of group G such that the order of c is 5 and the order of d is 3 respec-…

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Q: Consider the set of permutations V = {(1), (1 2) (3 4), (1 3) (2 4), (1 4) (2 3)}. Determine whether…

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Q: Prove that the following set form a group with addition as the operation: {0,5,10,15}

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Q: If A is an abelian group with A <G and B is any subgroup of G, prove that ANB < AB.

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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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Q: Verify that (ℤ, ⨀) is an infinite group, where ℤ is the set of integers and the binary operator ⨀ is…

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