Prove that every subgroup of Z is either the trivial group, {0}, or nZ = {nx | x E Z} for some n E N. %3D

Q: a(H n K) Let G is a group, H, K C G, and a e G. Is it the case that aH N aK? Provide a proof or…

A: According to the given information,

Q: The group generated by the cycle (1,2) is a normal subgroup of the symmetric group S3. True or…

A: Given, the symmetric group S3={I, (12),(23),(13),(123),(132)}. The group generated by the cycle (12)…

Q: (3) Let H be a subgroup of a group G. Let a E G. Prove that a¹Ha = {a¹ha he H} is a subgroup of G.

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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: (True/False) Suppose G is a group of order 25. Then every proper subgroup of G is abelian.…

A: (a) We know any group of order p2 is abelian where p is a prime number. Also, any subgroup of an…

Q: (b) Let G be a group and let x E G. Suppose that (x) is a cyclic group of order N. Let m and n be…

A: We make use of the gcd to answer this question. DEtails are given in step 2.

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

A: The given problem is related with group theory. G is a group and a ∈ G. Here, we write Cga instead…

Q: describe all the elements in the cycle subgroup GL(2,R) generated by the given 2 x 2 matrix [0 -1]…

A:

Q: Let G be a group and a e G. Prove that C(a) is a subgroup of G. Furthermore, prove that Z(G) = NaeG…

A:

Q: Let G be an abelian group and m be a positive integer. Then the set mG = {mn 1 x E G} and G(m) = {x…

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

Q: 5.3 Let H be the set of elements () of GL(2, R) such that ad-bc-1. Show that H is a subgroup of…

A: be the set of elements  of  such that  that is the We have to show that  is a subgroup of

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: Let G be a group, and let H < G. Assume that the number of elements in H is half of the number of…

A:

Q: Let G be a group with operation * and e be the identity element of G. Show that for any element a in…

A:

Q: (2) Let G be a finite group and H and K be subgroups of G such that HCKCG. Prove that [G: H] = [G:…

A: Let us prove this beautiful result using Lagrange's theorem. Lagrange's theorem = If G is a finite…

Q: Prove that H= { |ne Z} is a subgroup of GL2(R) under multiplication.

A:

Q: Let G Są and let K = {1,(1 2)(3 4), (1 3)(2 4), (1 4)(2 3)}. K is a normal subgroup of G. What is…

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Q: Let G be a group with identity e, and let a ∈ G. If a6 = e but a2 ≠ e, what can we say about |a|?…

A: Given that G be a group with identity e, and let a ∈ G. If a6 = e but a2 ≠ e.