For the group homomorphism : Zs → Z, defined by ø([r]) = [æ]² for all [r] € Z, find the kernel and image of ø.

Q: Let G be a group. Using only the definition of a group, prove that for each a E G, its inverse is…

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Q: Let G be the subgroup of GL3(Z₂) defined by the set 100 a 10 bc1 such that a, b, c Z₂. Show that G…

A: The given set of matrix is 100a10bc1 where a, b, c∈ℤ2. To find: the group to which the given set is…

Q: Let x be an element of group G. Prove that if abs(x) = n for some positive integer n, then x-1 =…

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Q: For each of the following values of n, describe all the abelian groups of order n, up to…

A: Given information n=10.

Q: For each of the following values of n, describe all the abelian groups of order n, up to…

A: For positive integer n, let Cn denote a cyclic group of order n. If G is an abelian group of order…

Q: Consider the map o : G¡ → G2 defined by: 9(a) = a-! %3D (a) Does o define group homomorphism? (b)…

A: Property of homomorphism of groups....

Q: For any group elements a and x, prove that |xax-1| = |a|.

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Q: Show that if H is any group and h is an element of H, with h" = 1, then there is a unique…

A: Given that H is a group and h ∈H  Now,we define a mapping f:Z→H  such that   f(n) = hn for n∈Z For…

Q: Let h: G G be a group homomorphism, and gEG is an element of order 35. Then the possible order of…

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

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Q: let x be an element of group g. Prove that if |x|=n then x^-1=x^n-1

A: Given 'x' be an element of a group G and |x|=n. As G be a group , inverse of each element of G must…

Q: Let G be the subgroup of GL3(Z2) defined by the set 100 a 10 b C 1 that a, b, c Z₂. Show that G is…

A: Given: G is the subgroup of GL3ℤ2 which is defined by the set of matrix 100a10bc1 where a, b, c∈ℤ2 .…

Q: Let a e G. Prove that $(a") = ¢(a)" for all n e Z.

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Q: Let U(n) be the group of units in Zn. If n > 2, prove that there is an element k EU (n) such that k2…

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Q: In Z, let A = <2> and B = <8>. Show that the group A/B is isomorphic to the group Z4 but…

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Q: for every element a of a ((a) Prove that group 6, 2 (G) is a subset of

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Q: Prove that if a is the only element of order 2 in a group, then a lies inthe center of the group.

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Q: Show that R* is isomorphic to G?  R* is a group under multiplication G is a group under addition…

A: To show A is one-one Let Ax1=Ax2 where x1 and x2 are two points of R*⇒x1-1=x2-1⇒x1=x2Thus the…

Q: Can you write a group homomorphism as φ (gh) as φ(hg)? Are they the same thing?

A: The given homomorphism ϕgh, ϕhg The objective is to find whether the ϕgh,ϕhg are same.

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

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Q: Let(G,*) and (H,#) be a groups if f: G H and g: H G are homomorphism such that gof = IG.f og = IH…

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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: If f: G to H is a surjective homomorphism of groups and G is abelian, prove that H is abelian.

A: As we know that a group homomorphism f:G to H is a map from G to H satisfying:

Q: Let G be a group with the order of G = pq, where p and q are prime. Prove that every proper subgroup…

A: Consider the provided question, Let G be a group with the order of G = pq, where p and q are prime.…

Q: For each of the following values of n, describe all the abelian groups of order n, up to…

A: Need to evaluate all the abelian group of order 36.

Q: Prove that each of the following sets, with the indicated operation, is an abelian group. (a) (R, *)…

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Q: (a) Prove that every element of Q/Z has finite order. (b) Given two groups (G, .) and (H, *).…

A: This question is related to group theory.

Q: Let Dg be the Dihedral group of order 8. Prove that Aut(D8) = D8.

A: We have to solve given problem:

Q: For each element a in the group G, define the mapping ka: G → G by ka(x) = xa-'for all x in G. %3D…

A: Since, you have posted multiple questions, but according to guidelines we will solve first question…

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: Let (G,*) be a group. Show that (G,*) is abelian iff (x * y)² = x² * y² for all x, y E G.

A: If a group G is abelian, then for any two elements x and y,    (x*y) = (y*x)   now associative…

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 G be a group, and a, b € G. Prove that b commutes with a if and only if b- commutes with a.

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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: Let H be the set of elements (ª of GL(2, R) such that ad– bc=1. Show that H is a subgroup of GL(2,…

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Q: For each of the following values of n, describe all the abelian groups of order n, up to…

A: We use the following result here. Result: Let n=pq for some prime numbers p and q with p<q. If p…

Q: 2. For an arbitrary set A, the power set P(A) = {X | X C A}, and addition in P(A) defined by X+ Y =…

A: Recall the following. Definition of group: Suppose the binary operation * is defined for element of…

Q: Let G be a finite group, prove that there exists m E G such that a ^ m = e for each a E G and where…

A: Let G be a finite group, prove that there exists m E G such that a ^ m = e for each a E G and where…

Q: Let G be a group and let x EG of order 23. Prove that there exist an element z EG such that z6 – x.

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Q: 4. Consider the additive group Z. Z Prove that nZ Zn for any neZ+.

A: We know that a group G is said to a cyclic group if there exists an element x of the group G such…

Q: Let x, y be elements in a group G. Prove that x^(−1). y^n. x = (x^(−1).yx)^n for all n ∈ Z.

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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: For the homomorphism : D4 → ({1, −1}, •) defined by Þ(a) = { 1 -1 if a is a rotation if a is a…

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Q: Prove that the subgroup {o E S | o (5) = 5} of Sg is isomorphic to S4.

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Q: Let F:Z → Z, be a mapping from the group of integers Z to a group of integers mod 6, Z, defined by:…

A: Ker(F) contains elements of Z whose image is zero under mapping F.

Q: 1. Consider the groups (R+, ) and (R,+). Then R* and R are isomorphic under the mapping $(x) = log10…

A: We use the definition of cosets, isomorphisms to answer these questions. The detailed answer well…

Q: Let G and H be groups, and let ø:G-> H be a group homomorphism. For xeG, prove that )x).

A: Given:G and H be the groups.

Q: On Z is defined the following binary operation; x . y = x + y - 1 Show that (Z, . ) is an abelian…

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Q: Let R be he set of real mumbers and let bER %3D 1 Show that G is abelian group an under…

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Q: 2. Let H and K be subgroups of the group G. (a) For x, y E G, define x ~ y if x = hyk for some h e H…

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