Let m and n be relatively prime positive integers. Prove that the order of the element (1, 1) of the group Zm x Zn is equal to mn. Conclude that the group Zm × Zn is isomorphic to Zmn-

Q: Prove that if x is a group element with infinite order, then x^m is not equal to x^n when m is not…

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Q: If g and h are elements from a group, prove that ΦgΦh = Φgh

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Q: Prove that the group G = [a, b

A: Given, the group G=a, b with the defining set of relations…

Q: Consider the group G = {x € R such that x # 0} under the binary operation x*y=-2xy The inverse…

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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: Suppose that for the three elements x, u, v of a group G, x=11v=vu, u²=e, v9=e where p and q are…

A: Given That : three element x, u ,v of a group G, x=uv=vu,up=e,vq=e where p and q are relative prime…

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

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Q: 12. Prove that the following groups are not cyclic: (a) Z2 x Z2 (b) Z2 x Z (c) Z x Z.

A: To show the following groups are not cyclic. If a group is cyclic the there exit an element in that…

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

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

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Q: Consider the group G = {x E R such that x 0} under the binary operation x*y=-2xy O x*x*x=4x^3…

A: Multiplication of the elements of the group elements with respect to binary operation

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: Suppose that the fundamental group of X is Z and p(xo) is finite. Find the fundamental group of X.

A: Given fundamental group of X is  ℤ and p-1(xo) = finite value Now we have to find the fundamental…

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

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Q: Suppose n km for positive integers k, m. In the additive group Z/nZ, prove that |[k],| = m, where…

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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: Prove that the group G with generators x, y, z and relations z' = z?, x² = x², y* = y? has order 1.

A: In order to solve this question we need to make the set of group G by finding x, y and z.

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: Consider the group G = {x € R such that x # 0} under the binary operation *: x*y=-2xy O x*x*x=4x^3 O…

A: Thanks for the question :)And your upvote will be really appreciable ;)

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

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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: 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 = {a +b 2 ∈ ℝ │ a, b ∈ ℚ }. Prove that the nonzero elements of G form a group under…

A: Note : In the question, it has to be a + b2 ∈ ℝ instead of a + b 2 ∈ ℝ. So, we are solving the…

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

A: Simple group of order 200

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

A: To prove that in a group (a-1 )-1=a for all a.

Q: Suppose that a is an element of a group G. Prove that if there is some integer n, n notequalto 0,…

A: Suppose that a is an element of a group G. Prove that if there is some integer n, such that n≠0 and…

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: Consider the group G= (x ER such that x ± 0} under the binary operation * x*y=-2y The inverse…

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Q: 3. Use the three Sylow Theorems to prove that no group of order 45 is simple.

A: Simple group: A group G is said to be simple group if it has no proper normal subgroup Note : A…

Q: Consider the group G = {x E R such that x # 0} under the binary operation ху X * y = 2 The identity…

A: First option is correct.

Q: (b) Given two groups (G,) and (H, +). Suppose that is a homomorphism of G onto H. For BH and A:{g…

A: Given that G,·and H,* are two groups.

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 = Z[i] = {a+bi | a, b € Z} be the Gaussian integers, which form a group under addition. Let y…

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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 = {2'3m5n : l, m, n E Z}. G is a group of rational numbers under the usual multiplication.…

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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: Find the group homomorphism between (Z, +) and (R- (0},.)

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Q: Consider the group G = {x € R such that x # 0} under the binary operation ху x* y = The order of the…

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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: 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 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 be an infinite cyclic group. Prove that G (Z,+)

A: To show that any infinite cyclic group is isomorphic to the additive group of integers

Q: Let G be a group, a E G. Prove that a=a + a < 2

A: Concept:

Q: Let G be a group and suppose that (ab)2 = a²b² for all a and b in G. Prove that G is an abelian…

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Q: according to exercise 27 of section 3.1 the nonzero elements of zn form a group g with respect to…

A: We know that a finite group is said to be cyclic if there exists an element of group such that order…

Q: Consider the group G = {x € R such that x # 0} under the binary operation *. ху X * y = x * 2 The…

A: First we have to find the identity element. Let G be the group and e be the identity element of G.…

Q: Let G, H, and K be finitely generated abelian groups. Prove or disprove: If G × H ∼= K × G, then H…

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Permutations and Combinations

If there are 5 dishes, they can be relished in any order at a time. In permutation, it should be in a particular order. In combination, the order does not matter. Take 3 letters a, b, and c. The possible ways of pairing any two letters are ab, bc, ac, ba, cb and ca. It is in a particular order. So, this can be called the permutation of a, b, and c. But if the order does not matter then ab is the same as ba. Similarly, bc is the same as cb and ac is the same as ca. Here the list has ab, bc, and ac alone. This can be called the combination of a, b, and c.

Counting Theory

The fundamental counting principle is a rule that is used to count the total number of possible outcomes in a given situation.