a. If G = {1,–1, i, -i}. In a group (G , ×) [Hint : i? = -1 or i = v-1] 1. Show that G is closed under x . 2. Show that (G , ×) is a cyclic group generated by i.

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…

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

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Q: 6. Show that for any two elements x, y of any group G, o(xy) = o(yx). %3D

A: Fact 1:In a group G, if x∈G such that xn = e, then O(x)|n (e→ identity element.)i.e. order of x…

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Q: prove that the group G=[a b] with defining set of relations a^3=e, b^7=e, a^-1ba=b^8 , is a cyclic…

A: We need to prove that , group G = a , b  with defining sets of relations a3 = e   ,  b7 = e  also…

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A: Given: The order of group is 15

Q: (H,*) is called a of (G,*) if (H,*) is a group.

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Q: Prove that E(n) = {(A, ¤) : A e O(n) and E R"} is a group. %3D

A: Consider the given: E(n)={(A,x)} where A∈O(n)and x∈ℝn

Q: One of the following is True O If aH = bH, then Ha = Hb. O Only subgroups of finite groups can have…

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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…

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Q: Let G be a group and let g, h ∈ G. Show that | gh | = | hg |. Remember that | a | denotes the order…

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Q: Let G be a group of odd order. Show that for all a E G there exists b E G such that a = b?.

A: Consider the given information, Let G be a group of odd order then, |G|=2k+1 where k belongs to…

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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: W6 Assume that H, k, and k are SubgrouPs of the group G and k, , Ka 4 G. if HA k, = HN k Prove that…

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Q: 1. Assume (X, o) and (Y, on X x Y as are groups. Let X × Y = {(x, y) |x € X,y E Y} and define the…

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Q: Show that ( Z,,+,) is a cyclic group generated by 3.

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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: Which of the following groups are cyclic? For each cyclic group, list all the generators of the…

A: To identify the given group is cyclic or not.

Q: Let G be a group and let a e G. In the special case when A= {a},we write Cda) instead of CG({a}) for…

A: Consider the provided question, According to you we have to solve only question (3). (3)

Q: 1- Prove that if (Q -(0),) is a group, and H = an, m e Z} 1+2m is a subset of Q - {0)}, then prove…

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Q: 1- Prove that if (Q - {0},) is a group, and H = 1+2n 1+2m 9 n, m e Z} is a subset of Q-{0}, then…

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Q: 1+2n Prove that if (Q-(0},) is a group, and H = a n, m e Z} 1+2m is a subset of Q-{0}, then prove…

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Q: Explain the following statement "If G is a group an a E G then o(a) = | |." 31. %3D

A: Order of element a ∈ G is the smallest positive integer n, such that an= e, where e denotes 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: 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: 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: Suppose H and K are subgroups of a group G.  If |H|=12 and |K| = 35, find |H intersected with K|.…

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Q: 8. Let (G,*) be a group, and let H, K be subgroups of G. Define H*K={h*k: he H, ke K}. Show that H*…

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Q: 4. Prove that the set H = nEZ is a cyclic subgroup of the group GL(2, R).

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

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Q: I'm unsure of how to approach this... Let N be a finite group and let H be a subgroup of N. If |H|…

A: According to the given information, Let N be a finite group and H be a subgroup of N.

Q: Show that the set S = (1, i, - 1, -0 is an abelian group with respect to the multiplication

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Q: Let (G,*) be a group such that a² = e for all a E G. Show that G is commutative.

A: A detailed solution is given below.

Q: 5) In each of parts (a) to (c) show that for specified group G and subgroup A of G, CG(A) = A and…

A: Given G=S3 and A=1,1 2 3, 1 3 2 The objective is to show that CGA=A, NGA=G. The definition for…

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Q: G, ba = ca implies b = c and ab = ac implies b = c for elements a, b, c E G. 31. Show that if a? = e…

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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: Let x belong to a group. If x2e while x : x + e and x + e. What can we say about the order of x? =…

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

A: Concept:

Q: F. Let a e G where G is a group. What shall you show to prove that a= q?

A: Solution: Given G is a group and a∈G is an element. Here a-1=q

Q: (d) Prove that 1+ N is a group with respect to multiplication. (e) Verify that 1+N(Z27) is a cyclic…

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Q: 7. You have previously proved that the intersection of two subgroups of a group G is always a sub-…

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Q: 2. If H and K are subgroups of an abelian group G, then HK = {hk | h e H and k e K} is a subgroup of…

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Q: Which of the following is cyclic group а. R b. Z С. Q d. C

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Q: For G = (Z5 ,+s) , how many generators of the cyclic group G? 5.a O 1.b O 3.c O 4.d O 2.e O

A: