Let G be a finite group. The center Z(G) of G is defined to be the set of all group elements z E G such that z.g = g·z for all g € G. Show that Z(G) is an abelian subgroup of G.

Q: Use Cayley's theorem to find an explicit subgroup of Sg which is isomorphic to the quaternion group…

A: Given Information:Group and Quaternion group .To find:The subgroup of  isomorphic to the Quaternion…

Q: Let G be an abelian group and let H= {g EG: g^n= e for some nEN}. Prove that H is a subgroup of G.

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Q: Let G = <x, y | x2n = e, xn = y2, y-1xy = x-1>. Show that Z(G) ={e, xn}. Assuming that |G| =…

A: To show that Z(G) = {e, xn}, we need to find all elements in G that commute with every other element…

Q: Suppose that g and h induce the same inner automorphism of a group G. Prove that h-'g E Z(G).

A: Given: g , h induce the same inner automorphism of a group G. Prove that: h-1g∈ZG.

Q: Prove that a group G is abelian if and only if the mapping f: G→G, given by f(x)=x-1,  is a…

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Q: Suppose that G is an Abelian group and o is an automorphism of G. Prove that H = {xE G|6(x) = x-1}…

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Q: Let f: G H be an injective homomorphism of groups. Prove G is abelian if H is abelian.

A: Proof given below. Explanation:

Q: Let G be a group. Prove that the mapping a(g) = g-1 for all g in G is an automorphism if and only if…

A: Introduction  A number transformation out of a class onto itself is called a group automorphism.…

Q: Let G be an abelian group and m be a  positive integer.  Then the set             mG={mn∣x∈G}  and…

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Q: If f is a homomorphism of a group G into a group G' with Kernal K. Let a € 6 be such that f(a) = a'…

A: We are given that f : G → G' is a homomorphism. Kernel of f is K.     => for all elements x in K,…

Q: 2. First Isomorphism Theorem. Let G be a group and H-((.s)|ge G). (a) (b) (1) G is abelian. (i) H is…

A: Given G is a group then, G×G is a group of the type such that G×G={(a,b); where a, b belong to G}…

Q: Let C(x) represent the centralizes of a. If x is an element of group G with the order of x be 5 (|x|…

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Q: Let G be a group and let a be an element of G. Prove that a func- tion fa : G → G defined by fa(g) =…

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Q: For any two elements a and b of a group G, show that G  is abelian iff…

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Q: Let G be a group and g E G. Show that Ø:Z→G by Ø(n) = g" is a homomorphism and isomorphism.

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Q: A homomorphism f of a group G into a group G' is an ismorphism iff ker f = {e}.

A: We have to proof that a homomorphism f of a group G into a group G' is an isomorphism if and only if…

Q: Let G be a group isomorphic to Zn, O Zn, O ...OZ. Let x be the product of all elements in G.…

A: Given: G is a group isomorphic to ℤn1⊕ℤn1⊕....⊕ℤnk and x is the product of all elements in G.To…

Q: Let G be an Abelian group. Determine all homomorphisms fromS3 to G.

A: Given: G is an Abelian group.To find: All homomorphisms from S3→G.

Q: Suppose G is a group and Z (G) and lnn (G) are the centers and groups of internal deformations of G,…

A: Let G is a group and Z (G) and lnn (G) are the centers and groups of internal deformations of G

Q: (14) Let G be a finite cyclic group of order n with identity element e. Prove that if a € G, then cd…

A: The order of every subgroup of a finite group G is divisor of the order of G.

Q: Let G be an Abelian group of order 16. Suppose that there are elementsa and b in G such that |a| =…

A: Given: G be an Abelian group of order 16 and a=b=4.To find: The isomorphism class of G.

Q: Let G be a finite group and let H be a normal Sylow p-subgroupof G. Show that a(H) = H for all…

A: Let α be an automorphism of G, and let p be a prime such that H is a Sylow p-subgroup of G.

Q: (B) Let G be a set of all real numbers excep: . Define * on G by a * b = |a|b, where |a| is the…

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Q: Let G be a group. Define ø : G → G by ø(x) = x-1 for all x E G. (a) Prove that ø is one-to-one and…

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Q: Assume that G is an abelian group. Let H C G be a subgroup. Prove that the factor group G/H is…

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Q: Let G be a finite abelian group

A: Let d = gcd(m; n).We shall use induction on d.Assume  first that d = 1.

Q: (c) Let G, H be groups (i) Ifo: G H is a homomorphism, then prove that o(2") = (o(x))" for all ne Z+

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Q: Let G be a group. The centre of G is defined as the set Z(G) = {g ∈ G : gx = xg for all x ∈ G}.…

A: Let G be a group. The center ZG of group G is defined as follows. ZG = g∈G : gx = xg , ∀x∈G We need…

Q: Show that any group of order 4 or less is abelian

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Q: Let (G, ★) and (G,o) be groups, with respective identity elements e and ē. Suppose there is a…

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Q: Let G be a group (not ncesssarily an Abelian group) of order 425. Prove that G must have an element…

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Q: f f is a homomorphism of group G into a  group G' with Kernal K, then K is a normal  subgroup of G.

A: If f is a homomorphism of group G into a group G' with Kernlal K, then we have to prove that K is a…

Q: 3 be group homomorphisms. Prove th. = ker(ø) C ker(ø o $).

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Q: Let (G,*) be a group such that x² = e for all x E G. Show that (G,*) is abelian. (Here x² means x *…

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Q: a) Check whether G = (3n, ·) is a group, where n ∈ Z. b) Let p1 (x) = x3 + 1 and p2 (x) = x4 + x.…

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Q: Let G be a group of order pq, where p and q are primes. Prove that either G is abelian or Z(G) =…

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Q: Let G be a group and let H₁, H₂ be subgroups of G such that H₁ H₂ and H₂H₁. (a) Prove that the union…

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Q: If G is a group of odd order and x ∈ G, show that x-1 is not in cl(x).

A: It is given that, G be a group of odd order and x∈G. Then, we have to show that x-1 is not in cl(x).…