Let G be a group. Prove the following. (a) For any i ∈ G, i and i's inverse have the same order. (b) a and xjx−1 have the same order for any j ∈ G. (c) For any a, b ∈ G, ab and ba have the same order, (ab not necessarily equal to ba).

Q: (h, k) E H x K. What is the order Let H and K be finite groups and let G = H x K. Let g of g?

A: To calculate the order of the element (h,k) in the direct product of groups

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Q: Consider the groups A = Z3 © Z9 and B = Z27. Prove that these two groups are not isomorphic.

A: The two groups G and G' are isomorphic iff there exists an onto isomorphism between them .

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Q: Prove that a group of order 28 must have an element of order 2.

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Q: let x and g be elements of a group G. show that x and gxg-1 have the same order. now prove that xy…

A: Hence, proved.Explanation:Step 1:To prove that x and gxg-1  have the same order for any elements x…

Q: (c) If K is a subgroup of G, then p(K) is a subgroup of H. Given: K < G (d) If K' is a subgroup of…

A: I have mentioned the test used to prove a subset to be a subgroup, as a note above. You can skip it…

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Q: Let a and b belong to a group.   If |a| = 10 and |b| = 21, show that n = {e}

A: Consider a group G. Let a and b be elements of the group G such that a=10 and b=21. Consider the…

Q: Prove that G = {a+b√2: a, b € Q and a and b are not both zero} is a subgroup of R* under the group…

A: Given- G=a+b2: a,b∈ℚ and a and b are not both zero To prove- G is a subgroup of ℝ* under the group…

Q: Write Q(sqrt2) for the set described in exercise 3.1 (iii). given a non zero element a + bsqrt(2) of…

A: It refers to the set Q(2), which is the set of all numbers of the form a+b2, where a and b are…

Q: 3 be group homomorphisms. Prove th. = ker(ø) C ker(ø o $).

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Q: 10. Let G = Z/36Z. (a) Find the order of every element of the group G. (b) Find all generators of G.

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

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Q: Determine which of the following are groups under the stated operations. For those which are groups,…

A: i) the pair here we draw composition table :246824826486426246886284from above table ,we concluded…