Let H x K = { (h,k) | h in H, k in K } such that (h1,k1) + (h2,k2) = (h1 + h2, k1 + k2), for all h1, h2 in H and k1, k2 in K. Now, if we have Z_2, then Z_2 x Z_2 is a O semigroup monoid O group O groupoid

Q: b). Let o:Z-Z be given by .0(n)=7n. Prove that o is a group homomorphism. Find the kernel and the…

A: As per our guidelines we are supposed to answer only one asked question.kindly repost the other…

Q: Let m, n ɛ Z+ such that gcd(m, n) = 1. Suppose there are M and N abelian groups, up to isomorphism,…

A: Given that m,n∈ℤ+ such that gcdm,n=1. Suppose that there are M and N abelian groups, up to…

Q: Exercise 7.2.10. Let G be a group of order pq for primes p and q. Prove all proper subgroups of G…

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Q: ii) Does there exist a group G such that G/[G,G] is non-abelian? Give an example, or prove

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Q: Show that SL(n, R) is a normal subgroup of GL(n, R). Further, by apply- ing Fundamental Theorem of…

A: Suppose, ϕ:GLn,R→R\0 such that ϕA=A for all A∈GLn,R Now, sinceA∈GLn,R if and only if A≠0 Now, we see…

Q: Theorem :- Let (G,-) be a group then :- 1- (Hom(G),) is semi group with identity. 2- (A(G),) is…

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Q: Let m, n e Zt such that gcd(m, n) = 1. Suppose there are M and N abelian groups, up to isomorphism,…

A: We have M and N are abelian groups of order m and n. so, we know that MN is abelian group.

Q: abelian group is not cyclic if and only if it contains a subgroup isomorphic to Zp×Zp

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Q: Q3\ Prove that if (G,*) be a finite group of prime order then (G,*) is an abelian group.

A: (G, *) be a finite group of prime order   To prove (G, *) is an abelian group

Q: (1) Z/12Z (2) (Zx 끄)/(6Zx 14Z) (3) (Z4 × Z4)/((2, 3)) (4) (Z4 x Z10)/((2, 4))

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Q: Exercise 2: Let G be a group and a€G. For any m, nƐZ, prove that a™+n = a"a" and (a")" = a™".

A: Note:   A non-empty  set  G  is  said  to be a groupwith  respect  to  a binary  composition  o  ifi…

Q: Let G be a group containing elements a and b. Express (ab)^2 without parentheses. Do not assume that…

A: Given that G is a group and let a,b belongs to G. The given expression is

Q: Let G be a group with the property that for any x, y, z in the group,xy = zx implies y = z. Prove…

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Q: Exercises 3(a) Prove that the groups $ and D6 are isomophic ; (e) Preve that the additive and the…

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Q: Let (G1, •) and (G2 , *) be two groups and o: G1→ G2 be an isomorphism. Then * O G2 might be abelian…

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Q: is] Let G and H be groups, and let T:G→H_be Isomorphism. Show that if G is abelian then H is also…

A: Note: We’ll answer the first question since the exact one wasn’t specified. Please submit a new…

Q: Let Z[x] be the group of polynomials in x with integer coefficientsunder addition. Prove that the…

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Q: 5/ Let G be group of class p9 a Prime Setting that proves that actual Subgroup of G is a cyclie is a

A: We know that every group of prime order is cyclic

Q: et G be a group with order n, with n > 2. Prove that G has an element of prime order.

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Q: 9. Classify the following groups in the sense of the Fundamental Theorem of Finitely Gener- ated…

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Q: 1. Show that H={[0], [2], [4]} is a subgroup of a group (Z6+6). Obtain all the distinct left cosets…

A: Given that H=0,2,4 and let G=ℤ6,+6.

Q: Let G = (-1,1) and (x,y) --> x ○ y = x + y/1 + xy is a binary operation on G. Let R+ = (0, ∞), with…

A: Given: G,O  with G=(-1, 1) ,  (x, y)→ x O y = x+y1+xy   and ℝ+, ·, ℝ+ = (0, ∞) with standard…

Q: Show that the units in Commutative ring unit element forms with an abelian group wv.t multipli…

A: let the two units of ring  R are u  and v let u-1 and v-1 also belong to R. therefore, uu-1=1 and…

Q: G group is called meta- Abelian if it has an Abelian subgroup N which is normal in G G and is…

A: (Solving the first question as per our guidelines) Question: Show that S3 is meta-Abelian.

Q: Q3\Prove that if (G,*) be a finite group of prime order then (G,*) is an abelian group.

A:

Q: 5. Prove that no group of order 96 is simple. 6. Prove that no group of order 160 is simple. 7. Show…

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Q: (1) Z/12Z

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Q: 17*. Find an explicit epimorphism from A5 onto a group of order 3

A: Epimorphism: A homomorphism which is surjective is called Epimorphism.

Q: By applying Fundamental Theorem of group homomorphism, show that the quotient groups GL(n, R)/SL(n,…

A: It is given that,  GL(n, ℝ) is the set of general linear group of all n×n matrices and  SL(n, ℝ) is…

Q: 6. Prove that if G is a group of order 231 and H€ Syl₁(G), then H≤ Z(G). me

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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: 5. Let p be a prime. Prove that the group (x, y|x' = yP = (xy)P = 1> is infinite if p> 2, but that…

A: The given group is <x,y  xp=yp=xyp=1>. To show: if p>2 then order of the given group is…

Q: a) Prove that, the characteristic of an integral domain R is either zero or a prime number. b) Let G…

A: this question related to ring theory and group theory.

Q: Let (G1, +) and (G2, +) be two subgroups of (R, +) so that Z+ ⊆ G1 ∩ G2. If φ : G1 → G2 is a group…

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Q: 6. Prove that if G is a group of order 231 and H€ Syl₁1(G), then H≤ Z(G). n Core

A: Given that,  G is group of order 231 and H∈syl11G. We first claim that there is a unique Sylow…

Q: (2) (Z x Z)/(6Z × 14Z)

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Q: Q3: (A) Prove that 1. There is no simple group of order 200. 2. Every group of index 2 is normal.

A: Sol1:- Let G be a group of order 200  i.e O(G) = 200                = 5² × 8. G contains k Sylows…

Q: (7) Define GL2 (R) to be the group of invertible 2 x 2 matric manifold, cc this group has the…

A: Define GL2R  to be the group of invertible 2×2 matrices.  To prove that this group has the structure…

Q: Prove that every finite Abelian group can be expressed as the (external) direct product of cyclic…

A: Fundamental Theorem of Finite Abelian Groups: Every finite Abelian group is a direct product of…

Q: 2) Prove that Zm × Zn is a cyclic group if and only if gcd(m, n) cyclic group Z; x Z4. = 1. Find all…

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Q: 2- Let (C,) be the group of non-zero -complex number and let H = {1,-1, i, -1}. Show that (H,;) is a…

A: We will be using definition of subgroup and verify that H indeed satisfy the definition.

Q: Prove that (Z × Z)/((0,1)) is an infinite cyclic group. Prove that (Z × Z)/((1,1)) is an infinite…

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Q: Show that Z12 is not isomorphic to Z2 ⊕ Z6.  ℤn denotes the abelian cyclic group of order n. Justify…

A: To show : ℤ12 is not isomorphic to ℤ2⊕ℤ6 Pre-requisite : P1. A group G is said to be cyclic if there…

Q: 5. Prove that the cyclic group Z/15Z is isomorphic to the product group Z/3Z × Z/5Z.

A: Definitions: Isomorphism: A mapping between two sets is called an isomorphism if it is one-to-one,…

Q: Let G be a group. V a, b, c d and x in G, if axb cxd then ab = cd then G is necessa: Abelian Of…

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Q: 9. Show that the two groups (R',+) and (R'- {0}, -) are not isomorphic. | 10. Prove that all finite…

A: Two groups G and G' are isomorphic i.e., G≃G′, if there exists an isomorphism from G to G'. In…

Q: Let (G,*) and (H,*) be finite abelian groups. If G x G = H x H then G=H. Show that they are…

A: Given that, G×G=H×H⇒G=H Since G,* and H,* are both finite abelian groups we get,…

Q: 34. Consider the action of S, the symmetric group of order 4, on Z[x1, X2, X3, X4] given by Let Hc…

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Q: Let G be a group. V a, b, c d and x in G, if axb = cxd then ab = cd then G is necessarily:…

A: The answer is given as follows :